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NUMERICAL METHODS FOR NONLINEARELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

TIAGO SALVADOR

Department of Mathematics and StatisticsFACULTY OF SCIENCE

McGill University, Montreal

MAY 2017

A thesis submitted to McGill Universityin partial fulfillment of the

requirements for the degree ofDoctor of Philosophy

© Tiago Salvador 2017

ABSTRACT

The goal of this thesis is to widen the class of provably convergent schemes for elliptic

partial differential equations (PDEs) and improve their accuracy. We accomplish this by

building on the theory of Barles and Souganidis, and its extension by Froese and Oberman

to build monotone and filtered schemes.

The first problem considered is the widely studied class of first order Hamilton-Jacobi

(HJ) equations. The goal is to construct provably convergent accurate schemes, together

with an efficient solver, by making use of the large number of discretizations and solvers

already available. To this end, we build filtered schemes, whose main idea is to blend a

stable monotone convergent scheme with an accurate scheme while retaining the advan-

tages of both: stability and convergence of the former, and higher accuracy of the latter.

Indeed, we are able to build schemes which are second, third, and fourth order accurate

in one dimension, as well as schemes that are second order accurate in two dimensions.

Moreover, the schemes are explicit, allowing us to use the easy-to-implement fast sweeping

method. Using different accurate schemes (e.g. from standard centred differences, higher

order upwinding and ENO interpolation), the accuracy of the filtered schemes is validated

with computational results for the eikonal equation, as well as other HJ equations (both in

one and two dimensions).

The second problem considered is the 2-Hessian equation, a fully nonlinear PDE related

to the intrinsic curvature for three-dimensional manifolds. The goal is to build numerical

methods to compute its solution on a bounded domain given prescribed boundary data.

We propose two distinct methods. The first is provably convergent to the unique viscosity

solution. The second has higher accuracy and converges in practice, but lacks a formal

proof of convergence. The PDE is elliptic on a restricted set of functions: a convexity-

type constraint is needed for the ellipticity of the PDE operator, which poses additional

difficulties when building the numerical methods. Solutions with both discretizations are

obtained using Newton’s method. Computational results are presented on a number of

exact solutions which range in regularity from smooth to non-differentiable, and in shape

from convex to non-convex.

The third and last problem is to build a provably convergent scheme for the nonlinear

PDE that governs the motion of level sets by affine curvature. It is closely related to mean

curvature but exhibits instabilities not found in the former. These instabilities and the lack

ii

Abstract iii

of regularity of the affine curvature operator posed unexpected and additional difficulties

in building monotone schemes. A standard finite difference scheme is proposed and an

example that illustrates its nonlinear instability is given. We build provably convergent

monotone finite difference schemes. Numerical experiments demonstrate the accuracy and

stability of the discretization, as well as the fact that our approximate solutions capture the

affine invariance and morphological properties of the evolution.

ABRÉGÉ

L’objectif de cette thèse est d’élargir la classe de schémas à convergence prouvable pour les

équations aux dérivées partielles (EDPs) elliptiques, et d’améliorer leur précision. Nous

l’accomplissons en nous appuyant sur la théorie de Barles et Souganidis, et l’extension de

celle-ci par Froese et Oberman pour construire des schémas monotones et filtrés.

Le premier problème considéré est la classe largement étudiée des équations de

Hamilton-Jacobi (HJ) du premier ordre. L’objectif est de construire des schémas pré-

cis et convergents avec un solveur efficace, en utilisant le grand nombre de discrétisations

et de solveurs déjà disponibles. À cette fin, nous construisons des schémas filtrés, dont

l’idée principale est de combiner un schéma monotone, convergent, et stable, et un schéma

précis, tout en conservant les avantages des deux. On préserve la stabilité et la convergence

du premier et la plus grande précision du deuxième. En effet, nous sommes en mesure de

construire des schémas de deuxième, troisième et quatrième ordre précis en une dimension,

ainsi que des schémas de deuxième ordre précis en deux dimensions. En outre, les schémas

sont explicites, ce qui nous permet d’utiliser la méthode dite de “Fast Marching”, qui est

facile à mettre en oeuvre. En utilisant différents schémas précis (par exemple, à partir de

différences centrées standard, de “upwinding” à ordre supérieur, et d’interpolation ENO),

la précision des schémas filtrés est validée avec des résultats informatiques pour l’équation

eikonal et d’autres équations de HJ (aussi bien en une qu’en deux dimensions).

Le deuxième problème considéré est l’équation 2-Hessienne, une EDP entièrement

non linéaire associée à la courbure intrinsèque sur des les variétés tridimensionnelles.

L’objectif est de construire des méthodes numériques pour en calculer la solution sur

un domaine borné, et avec les données à la frontière prescrites. Nous proposons deux

méthodes distinctes. On prouve que la première converge à la solution de viscosité

unique. Le seconde a une plus grande précision et converge en pratique, mais on manque

d’une preuve formelle de convergence. L’EDP est elliptique sur un ensemble restreint de

fonctions: une contrainte de type convexe est nécessaire pour l’ellipticité de l’opérateur

PDE, ce qui pose des difficultés supplémentaires lors de la construction des méthodes

numériques. Les solutions avec les deux discrétisations sont obtenues en utilisant la

méthode de Newton. Des résultats numériques sont présentés sur un certain nombre de

solutions exactes qui varient en régularité, de lisses à non différentiables, et en forme, de

convexes à non convexes.

iv

Abrégé v

Le troisième et dernier problème consiste à construire un schéma convergent pour

l’EDP non linéaire régissant le mouvement des ensembles de niveau par courbure affine.

Il est étroitement lié au problème de courbure moyenne, mais présente des instabilités

non trouvées dans ce-dernier. Ces instabilités, ainsi que le manque de régularité de

l’opérateur de courbure affine, posent des difficultés inattendues et supplémentaires

dans la construction de schémas monotones. Un schéma aux différences finies standard

est proposé, et on présente un exemple qui illustre son instabilité non linéaire. Nous

construisons un schéma aux différences finies monotones, dont on prouve la convergence.

Des essais numériques démontrent la précision et la stabilité de la discrétisation, ainsi

que le fait que nos solutions approximatives captent l’invariance affine et les propriétés

morphologiques de l’évolution.

STATEMENT OF CONTRIBUTION

Here we summarize the contributions of each chapter of this thesis, which we elaborate

more on each chapter.

Chapter 4 was published in the Journal of Computational Physics where my advisor

Prof. Adam Oberman is a coauthor [OS15]. He directed the research and I conducted the

research. The contributions of this chapter are:

• To build provably convergent accurate schemes for first order Hamilton-Jacobi equa-

tions.

Chapter 5 was published in IMA Journal of Numerical Analysis, where my advisor

Prof. Adam Oberman and Prof. Froese are coauthors [FOS16]. They directed the research

and I conducted the research. The contributions of this chapter are:

• To provide an accurate numerical method to compute solutions of the 2-Hessian

equation.

• To build a provably convergent monotone scheme for the same PDE.

Chapter 6 has been submitted for publication and is currently under review. My advisor

Prof. Adam Oberman is a coauthor [OS15]. He directed the research and I conducted the

research. The contributions of this chapter are:

• To study the nonlinearly instabilities exhibit by a standard finite difference scheme

for the PDE that governs the motion of level sets under affine curvature.

• To build a provably convergent monotone scheme for the same PDE.

vi

LIST OF FIGURES

3.1 Examples of filter functions: discontinuous filter (left), continuous filter (right). 36

4.1 Profile of the solutions of the five examples considered in one dimension (at the

top, eikonal equation examples, at the bottom, HJ equations examples). . . . . . 53

4.2 Exact solution and solutions obtained with the monotone scheme and the 2nd

order upwind filtered scheme with 50 grid points for the first example of the

eikonal equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3 Active stencils in the accurate scheme in the last iteration for the solutions of

the second example considered: −i means that i points to the left were used in

the interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.4 Log-log plot of the errors for the one-dimensional examples of the eikonal

equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.5 Log-log plot of the errors for the one-dimensional examples of HJ equations. . . 59

4.6 Profile and contour plots of the solutions of the three examples considered in

two dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.7 Log-log plot of the errors for the two-dimensional examples in the l∞ norm. . . 64

4.8 Log-log plot of the errors for the two-dimensional examples in the l1 norm. . . 64

4.9 Log-log plot of the errors for the second example in the l∞ norm in regions

(x, y) ∈ R2 : |x + y| > 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.10 Log-log plot of the errors for the third example in the l∞ norm in regions

(x, y) ∈ R2 : x2 + y2 ≥ 1, x ≥ 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.1 Failure of the parabolic solver using the naive finite difference scheme: section

z = 0.9 of the initial guess (left) and the solution after 25 iterations (right). . . . 76

5.2 Elements of V1 (solid) and elements of V2 \ V1 (dashed). . . . . . . . . . . . . . . 85

5.3 Surface plots of the level sets of the solution to Example 5.7 on a 30 × 30 × 30

grid with the naive finite differences (left) and the 27-point monotone scheme

(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.4 Plot of the curves t 7→ u(t, t, t) of the solution of Example 5.7 on a 30 × 30 × 30

grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

vii

viii LIST OF FIGURES

5.5 Surface plots of the level sets of the solution to Example 5.8 on a 30 × 30 × 30

grid with the naive finite differences (left) and the 27-point monotone scheme

(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.6 Plot of the curves t 7→ u(t, t, t) of the solution of Example 5.8 on a 30 × 30 × 30

grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.1 Solution of the one-dimensional model equation using standard finite differ-

ences at times t ∈ 0, 1, 2, 5. Here dt = h2/2 on a 256-point grid. . . . . . . . . . 114

6.2 Plot of the solution obtained described in Example 6.1 at time t ∈ 0, 1, 5, 20on a 128-point grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.3 Lack of convergence when using a standard finite difference scheme: example

6.6 (d) with the standard finite difference solver: level sets of the solution at

times t = 0 (upper left), t = 15 (upper center), t = 17 (upper right), t = 20

(lower left), t = 40 (lower center) and t = 50 (lower right) with dt = h2/2 on a

32 × 32 grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.4 Neighbour points of the stencil for nθ = 3 (smaller circle) and nθ = 7 (larger

circle). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.5 (Top:) Evolution of an ellipse by (left) affine curvature, (right) mean curvature.

(Bottom:) Evolution by affine curvature of (left) a diamond and (right) a flatter

diamond. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.6 Evolution of a fan-shape like curve under affine curvature motion (top) and

mean curvature (bottom) at time t ∈ 0, 0.05, 0.1, .2 (see Example 6.5 for more

details). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.7 Plot of the zero level sets in Example 6.10 of Φt(u φ) and(

Φt(det φ)2/3(u))

φ

for regular elliptic scheme (left), standard scheme (center) and regular filtered

scheme (right) at time t = 1 with φ given by (a) (top) and (b) (bottom). . . . . . 134

LIST OF TABLES

4.1 Accuracy in the l∞ norm and order of convergence of the schemes for the first

example of the eikonal equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Accuracy in the l∞ norm and order of convergence of the schemes for the

second example of the eikonal equation. . . . . . . . . . . . . . . . . . . . . . . . 56

4.3 Accuracy in the l∞ norm and order of convergence of the schemes for the third

example of the eikonal equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4 Accuracy in the l∞ norm and order of convergence of the schemes for the fourth

example (H(p) = p2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.5 Accuracy in the l∞ norm and order of convergence of the schemes for the fifth

example (H(p) = cos(p)2 + |p|). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.6 Accuracy and order of convergence of the schemes for the first example in two

dimensions in the l∞ norm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.7 Accuracy and order of convergence of the schemes for the second example in

two dimensions in the l∞ norm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.8 Accuracy and order of convergence of the schemes for the second example in

two dimensions in the l1 norm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.9 Accuracy and order of convergence of the schemes for the third example in two

dimensions in the l∞ norm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.1 Elements of G1 up to permutations. . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.2 nS is the number of ν directions available in the stencil, i.e., nS = #Vnθ. . . . . 84

5.3 Accuracy in the l∞ norm of the schemes for the first example using the Newton

solver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.4 Accuracy in the l∞ norm and order of convergence of the schemes for the

second example using the Newton solver. . . . . . . . . . . . . . . . . . . . . . . 91

5.5 Accuracy in the l∞ norm and order of convergence of the schemes for the third

example using the Newton solver. . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.6 Accuracy in the l∞ norm and order of convergence of the schemes for the fourth


ix

x LIST OF TABLES

5.7 Accuracy in the l∞ norm and order of convergence of the schemes for the fifth


5.8 Accuracy in the l∞ norm and order of convergence of the schemes for the sixth


6.1 Coordinates of the neighbours in the first quadrant of a stencil with width nθ. . 118

6.2 Accuracy in the l∞ norm and order of convergence of the schemes for Example

6.6 with regularized schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.3 Error in the l∞ norm of the whole computational domain at time t = 0.1 for the

time dependent Example 6.7 and 6.8. . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.4 Difference in the l∞ norm between Φt(gv u0) and gv Φt(u0) for v = 1, 2 for

Example 6.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

CONTENTS

Abstract ii

Abrégé iv

Statement of contribution vi

List of Figures vii

List of Tables ix

Acknowledgements xiii

1 Introduction 1

1.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Viscosity Solutions 11

2.1 Motivation of viscosity solutions . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Definition of viscosity solution . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Comparison principle and uniqueness . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Existence of viscosity solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Numerical Schemes 21

3.1 Monotone finite difference schemes . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Elliptic finite difference schemes . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Building Elliptic Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 Filtered finite difference schemes . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Filtered Schemes for Hamilton-Jacobi equations 39

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Discretization and solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

xi

xii CONTENTS

5 2-Hessian equation 67

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2 Background on the equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.3 Discretization and solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.4 Computational results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6 Affine curvature 99

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.2 The affine curvature PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.3 Numerical methods for the model equation . . . . . . . . . . . . . . . . . . . 105

6.4 Nonconvergence of standard finite differences . . . . . . . . . . . . . . . . . 113

6.5 Convergent finite difference methods . . . . . . . . . . . . . . . . . . . . . . . 117

6.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7 Conclusions 137

Bibliography 139

ACKNOWLEDGEMENTS

First and foremost I would like to thank my supervisor Professor Adam Oberman for

his guidance and support during my PhD. Without him this work would not have been

possible. I would also like to thank Professor Brittany Froese who I coauthor a manuscript

with and had many fruitful discussions. The staff in the Mathematics department were

always there to help me with any bureaucratic issues and deserve my many thanks. I

am also grateful for the funding I received that made this work possible, including the

scholarship from FCT, the Portuguese national funding agency for science, research and

technology.

I spent five amazing years at McGill and they will definitely be marked by the people I

met and became friends with. I was lucky enough to have amazing officemates: Bilal, Eric,

Chris, Geoff, were always available to discuss work and bounce off ideas or simply take a

break. A special thanks to Alexandra Tcheng for many lengthy and helpful discussions.

Daphna Harel and Sanam Joon thank you for your friendship, support and advice. To

all the friends I made at intramural soccer at McGill, playing with you was a pleasure

and provided the perfect escape from mathematics, in particular 2v2 soccer with Adam

Alcolado. Allison Kolly, thank you for you love and support, this last year and half would

have been much tougher without you.

There are two people who I met only once I arrived to Montreal and to who I am

more than grateful for all they did for me: Adriana Simões and Francisco Salvador. They

welcomed me in Montreal like family and made me feel like I was at home. One of the

reasons moving to Montreal was so easy was definitely because of them and that is why I

will be eternally grateful and forever in their debt.

Last but not least, I would like to thank my parents. Despite being on the other side of

the ocean, their love, support and advice was and will always be invaluable to me.

xiii

CHAPTER 1

INTRODUCTION

In this chapter, we discuss the three distinct elliptic partial differential equations (PDEs)

that are the subject of this thesis in Chapters 4, 5 and 6: Hamilton-Jacobi (HJ) equations

(with special focus on the eikonal equation), 2-Hessian equation and the PDE that governs

the motion of level sets by affine curvature. A literature review on each problem is

presented. We conclude the chapter with a description of the organization of the thesis.

Monotone schemes, introduced by Barles and Souganidis in [BS91], constitute the ideal

framework to build provably convergent schemes for elliptic PDEs. In a nutshell, the

theory says that monotone, stable, consistent schemes converge to the unique viscosity

solution of an elliptic PDE. This result makes monotone schemes very desirable. For

instance, they have been highly successful in building convergent schemes for the Monge-

Ampère equation, including the case of Optimal Transportation [BFO14], which suggest

they could be extended to more general geometric and optimal transportation type PDEs.

Despite the clear requirements of the theory, building monotone schemes is a challenge

since the definition does not provide any insight on how to build such schemes. Moreover,

conditions to ensure monotonicity are different for first and second order equations, and

for explicit and implicit schemes. Using the related notion of elliptic schemes, the work in

[Obe06a] provides a unifying and convenient reformulation of monotone schemes.

Another issue that arises when dealing with monotone schemes is the question of

accuracy. Unfortunately, the accuracy of monotone schemes is less than ideal in some

situations: it is at most first (resp. second) order for first (resp. second) order equations.

To address this, filtered schemes were introduced in [FO13] in the context of the Monge-

Ampère equation, where the goal was to overcome the reduction in accuracy based on the

use of a wide-stencil monotone scheme.

The main goal of this thesis, building on the foundation of monotone schemes, elliptic

and filtered schemes, is thus in widening the class of equations covered and extending the

theory and accuracy of the methods.

1.1 L ITERATURE REVI EW

In this section, we give an overview of some relevant existing literature on the three distinct

problems studied in this thesis.

1

2 1.1. LITERATURE REVIEW

1.1.1 Hamilton-Jacobi equations

In this thesis, we are interested in HJ equations of the form

H(x, ∇u) = f(x), x ∈ Ω,

u(x) = g(x), x ∈ Γ,

where ∇u is the gradient of the function u, Ω is an open set, Γ is the boundary of Ω and the

Hamiltonian H is a nonlinear Lipschitz continuous function. HJ equations appear in many

applications, such as optimal control, differential games, image processing, computer

vision and geometric optics.

We take a particular interest on the special case (take H(p) = |p|) of the eikonal equation

|∇u(x)| = f(x), for x outside Γ,

u(x) = g(x), for x on Γ.

where f > 0 and Γ is here a closed, bounded set. The eikonal equation has a wide range

of applications in geometric optics, computer vision, optimal control, etc. Moreover,

as pointed out in [BLZ10], high order schemes are particularly important in the high

frequency wave propagation where the eikonal equation is coupled to a transport equation

through its gradient [QS99, SVST94].

There are already a large number of discretizations and solvers available for HJ equa-

tions. The simplest approximations are finite difference schemes based on a Cartesian

grid. In this class, monotone schemes are provably convergent [BS91], but only first order

accurate [Obe06b]. In general, higher order finite difference schemes for HJ equations are

neither monotone, nor stable. For example, the centered difference scheme is unstable for

the eikonal equation [Set99b, Section 4.3]. The filtered schemes presented in this thesis are

designed to remain stable while allowing for a wide choice of accurate discretizations.

Higher order accurate schemes have been built, but only by giving up other desirable

properties (e.g. ease of implementation, fast solvers, or the convergence proof). Semi-

Lagrangian schemes [FF02, CF07], are accurate, but they involve solving the characteristic

ordinary differential equations, and are generally more complicated to implement. Central

schemes [LT00] achieve second order accuracy, at the expense of a slightly more compli-

cated, non-explicit formulation. The ENO and WENO schemes [OS91, Shu07, JP00] are

accurate, and while not provably convergent, they are effective in practice. Combinations

of WENO and central schemes have been implemented, achieving higher order accuracy

[BL03]. The ENO based schemes use adaptive stencils, which complicates the use of fast

solvers (however see [ZZQ06] for a sweeping method). Fast marching methods require

Chapter 1. Introduction 3

specialized data structures to implement, are usually first order accurate (however see

[ABM+11] for higher order methods) and only apply to the eikonal equation. A compact

upwind second order scheme for the eikonal equation was proposed in [BLZ10]. In the

same spirit of the filtered schemes presented here, a higher order scheme for HJ equations

was presented by Abgrall in [Abg09]. The convergence of this scheme also follows from

an adaptation of the Barles-Souganidis convergence proof.

1.1.2 2-Hessian equation

The second equation studied in this thesis is the 2-Hessian equation in three dimensions, a

fully nonlinear PDE of the form

S2[u] ≡ uxxuyy + uxxuzz + uyyuzz − u2xy − u2

xz − u2yz = f.

This is a particular instance of a much larger k-Hessian family of PDEs in n-dimensional

space, that include the Laplace equation, ∆u = f , when k = 1, and the Monge-Ampère

equation det D2u = f , when k = n.

Geometric PDEs have been proven to be especially useful in image analysis [Sap06].

In particular, the Monge-Ampère equation in the context of Optimal Transportation has

been used in three dimensional volume based image registration [HZTA04]. While the

2-Hessian equation is unfamiliar outside of Riemannian geometry and elliptic regularity

theory, it is closely related to the scalar curvature operator, which provides an intrinsic

curvature for a three dimensional manifold. Thus, one would expect that scalar curvature

equations would have been used in these contexts, which is not the case. Reasons for this

may include a lack of effective PDE solvers for this operator. Indeed, there are very few

publications devoted to solving the 2-Hessian equation. In the early work of [SG10] a

quadratically constrained eigenvalue minimization problem is solved. In the unpublished

work of [Awa14], an iterative method with quadratic convergence rate is proposed. Gauss-

Seidel and semi-implicit solvers, that relate to the ones we present here, are also discussed.

The 2-Hessian operator also appears in conformal mapping problems. Conformal

surface mappings have been used for two dimensional image registration [AHTK99,

GWC+04], but they do not generalize directly to three dimensions. Quasi-conformal maps

have been used in three dimensions [WWJ+07, ZG11], however these methods are still

being developed.

More generally, k-Hessian equations appear in some problems in differential geometry:

generalizations of the Yamabe problem [Via00b, Via00a], the Calabi-Yau problem [AV10]

and the Christoffel-Minkowski problem [GM03].


Scalar curvature and the 2-Hessian equation

The 2-Hessian equation corresponds to scalar curvature, as we discuss below, and solving

the 2-Hessian PDE (or a related one) allows for the construction of hyper-surfaces of

prescribed curvatures, for example scalar curvature [GG02]. We start by briefly discussing

the Gaussian curvature due to its relation to the Monge-Ampère equation.

The Gaussian curvature of a two-dimensional surface is the product of the principal

curvatures, κ1, κ2 of the surface. It is an intrinsic quantity: it does not depend on the

embedding of the surface in space. Locally, the surface can be defined as the graph of a

function u(x), whose gradient of the function vanishes at x. Then the Gaussian curvature

at x is given by the determinant of the Hessian of u(x), det(D2u) = κ1κ2, which is the two

dimensional Monge-Ampère operator applied to u (if the gradient of u does not vanish at

x, additional first order terms appear).

In higher dimensions, curvature is a tensor rather than a scalar quantity. The curvature

tensor is defined by the sectional curvature, K(p, x), which is given by the Gaussian

curvature of the geodesic surface defined by the tangent plane, p, at x. The scalar curvature

(or the Ricci scalar), which is the trace of the curvature tensor, is the simplest curvature

invariant of a Riemannian manifold. It can be characterized as a multiple of the average

of the sectional curvatures. If we choose coordinates so that a three dimensional surface

is given by the graph of a function u(x) whose gradient vanishes at x, then the scalar

curvature is given by the 2-Hessian operator:

1

2

(

trace(D2u)2 − trace(

(D2u)2))

= κ1κ2 + κ1κ3 + κ2κ3

where κ1, κ2, κ3 are the three principal curvatures. Again, if the gradient of u does not

vanish at x, additional first order terms appear. However the equation above holds in

general if we replace the principal curvatures with the eigenvalues of the Hessian, which

leads to the 2-Hessian equation.

Since the second order terms pose the primary challenge in the solution of nonlinear

elliptic equations, we focus on the 2-Hessian equation in this work. In a similar way, the

Monge-Ampère equation can be related to the equation for Gauss curvature through the

inclusion of appropriate first order terms. In [BFO14] an extension of the Monge-Ampère

equation with first order nonlinear terms was studied and the primary challenge was the

boundary conditions.


Related work on curvature equations

The 2-Hessian equation is closely related to a curvature PDE in three dimensions. In two

dimensions there are several works on the evolution of curves using curvature, going

back to the seminal paper of Osher and Sethian [OS88]. In [Obe04], a finite difference

monotone scheme is given for the motion of level sets by mean curvature. The advantage

of monotone discretizations is that they have a convergence proof, and convergent schemes

are more stable and allow for faster solvers [Set96]. The surface evolver [Bra92] is a tool to

evolve two dimensional surfaces by curvature based on the minimization of its energy. In

[Sap06] one can find a relation between geometric PDEs and image analysis. For a review

of the numerical methods for curvature flows see [DDE05a].

Related work on the Monge-Ampère equation

In this thesis we study a fully nonlinear elliptic PDE, while most of the curvature flows

lead to quasilinear parabolic equations. Thus, we also review some of the related work on

the Monge-Ampère equation, a fully nonlinear elliptic PDE. For an extended review on

numerical methods for fully nonlinear elliptic PDEs see [FGN13].

The Monge-Ampère equation has been exhaustively studied. Consistent schemes

using either finite elements [Nei13, BN12] or finite differences [LR05] have been proposed.

However, these schemes are not monotone and therefore do not fall within the convergence

framework of Barles and Souganidis [BS91]. They require instead the PDE solution to be

sufficiently smooth and the numerical solver to be well initialized. Using wide stencil

discretizations, consistent monotone schemes were built [FO11a, FO11b], which are thus

provably convergent but have limited accuracy due to their directional resolution. This

limitation has been overcome recently. By introducing filtered schemes, which blend a

monotone scheme with an accurate (but possibly unstable) scheme, the authors in [FO13]

were able to obtain a provably convergent scheme with improved accuracy. Two other

solutions, specific to particular dimensions, have been proposed as well: in the two

dimensional setting using a mixture of finite differences and ideas from discrete geometry

[BCM14] and in the three dimensional setting using ideas from discretizations of optimal

transport based on power diagrams [Mir15].

The Monge-Ampère problem is related to the problem of prescribed Gauss curvature as

already mentioned. Numerical methods for the problem of prescribed Gauss curvature can

be found in [MO16, Fro16a] . The Gauss curvature flow is also used in image processing

for surface fairing [EE09].


1.1.3 Affine curvature

The planar motion of level sets by affine curvature is governed by the nonlinear PDE

ut = |∇u| (k[u])1/3 . (AC)

Here u = u(x, y) : R2 → R, ∇u = (ux, uy) denotes the gradient of u, and k[u] denotes the

curvature of the level set of u

k[u] = div

(

∇u

|∇u|

)

=uxxu2

y − 2uxuyuxy + uyyu2x

(u2x + u2

y)3/2. (1.1)

The affine curvature PDE is closely related to the well known PDE for motion of level sets

by mean curvature

ut = ∆1u := |∇u| k[u]. (MC)

The affine curvature evolution is one of the most fundamental geometric evolution

equations, after the mean curvature evolution. It was introduced by Sapiro and Tannen-

baum in [ST94] and [AST98] and has applications in mathematical morphology, edge

detection, image smoothing, and image enhancement (see [Sap06]).

The study of the affine curvature PDE is motivated by the recent work of Jeff Calder

[CS16], which provides an application to the statistics of large data sets. The convex

hull peeling algorithm [Cha85] provides an affine invariant notion of the median and the

quantiles of multidimensional probability distributions. While there is more than one

way to measure data depth [Bar76], affine invariance is an important property for such

measures [LPS+99]. The level sets of the solution of the affine curvature PDE, with right

hand side given by the probability density ρ, also give an affine invariant notion of the

depth of ρ. According to [CS16], these two notions of depth are equivalent: the rescaled

data depth layers of N data points sampled from the density, ρ, given by convex hull

peeling algorithm converge, in the limit N → ∞, to the levels given by the solution of

the PDE. Compared to convex hull peeling, the PDE characterization is efficient in terms

of the number of data points N : an efficient density estimation method can be used to

approximate ρ, and afterwards the PDE solver does not depend on N . This kind of limiting

PDE approach has already been shown to be effective for non-dominated sorting [CEH14].

Euclidean and Affine Curvature

We begin with a parametric description of the affine curvature evolution, and make a

comparison with the more familiar mean curvature evolution. We refer to [Sap06, Chapter

2] for more details. Consider a curve described parametrically C(s) : [a, b] ∈ R → R2


where s parameterizes the curve. If the curve is parameterized by the Euclidean arc length,∣

∣

∣

dCds

∣

∣

∣ = 1, then the curvature, k, is defined, up to a sign, by |k| := |Css|. It follows then

that circles have constant mean curvature. Letting t and n denote respectively the unit

Euclidean tangent and the Euclidean normal of the curve, we have

dCds

= t andd2Cds2

= kn.

The affine curvature arises from a different parameterization of the curve. Define the

parameter, r by the condition that the vectors Cr and Crr form a parallelogram of area 1,

[Cr, Crr] = 1.

Here the brackets denote the determinant of the matrix whose columns are given by those

vectors. By differentiating the last equation, we obtain [Cr, Crrr] = 0, which implies that

Crrr = µCr for some constant µ. Using the defining condition again, we obtain

µ = [Crr, Crrr],

which we define to be the affine curvature of the curve. The affine curvature is the simplest

nontrivial affine invariant of the curve [Su83]. Ellipses have constant affine curvature.

Similarly to the Euclidean curvature, we can define the affine tangent vector and the affine

normal vector, which we denote by ta and na, respectively. The following relation then

holds:

na = k1/3n + f(k, kp)t,

where f is a function of the Euclidean curvature and its first derivative.

Under the affine curvature evolution, any convex curve remains convex; any convex

smooth curve evolves to an ellipse until it collapses to a single point; any smooth curve

becomes convex after a certain time. Moreover, the affine curvature evolution is invariant

under the class of special affine transformations, which are defined by matrices with

determinant 1. Compare this to the mean curvature evolution, which shrinks curves to

circles [Gag84] and is invariant under the smaller class of orthogonal transformations.

Level Set PDE formulation

The Level Set Method [Set99c, OF03] for the affine curvature evolution results in the

PDE (AC). It has the following advantages compared to parameterized curve evolution:

(i) it provides a natural generalization of the flows when the curve becomes singular and

notions such as normals are not well defined; (ii) there is no need to track topological


changes since they are discovered when the corresponding level set is computed; (iii) it

can be discretized on a uniform grid, which is convenient for many applications.

In the Level Set Method, a curve is represented implicitly as the level set of the auxiliary

function u(x, y, t) : R2 × R → R, that is,

C(t) =

(x, y) ∈ R2 | u(x, y, t) = c

for some arbitrary constant c ∈ R. If u satisfies ut = |∇u| β[u(·, t)], for some function β

which depends on the level set of u, then all its level sets move in the normal direction with

speed β. For example, choosing β = 1, we obtain the time-dependent eikonal equation,

ut = |∇u|. Taking β = k[u] or (k[u])1/3 we obtain (MC) and (AC), respectively.

Geometric curve evolution formulation

Equivalently, we can also think of the evolution of a single curve. Denoting by C(p, t) :

S1 × [0, T ) → R2 a family of closed curves, they move with β velocity in the normal

direction if the following equation is satisfied:

∂C(p, t)

∂t= β(p, t)n(p, t), (1.2)

with C0 as the initial condition. A more general formulation may allow for a tangential

component of the velocity. However, if β is a geometric intrinsic characteristic of the curve

(meaning it does not depend on its the parameterization), then the tangential component

in the velocity does not influence the geometry of the deformation of the curve, just its

parameterization. Taking β = k(p, t) or (k(p, t))1/3 we recover the evolution of a single

curve by mean curvature and affine curvature, respectively. Notice that both the evolutions

can be written as∂C(p, t)

∂t= Css

where s denotes the Euclidean arc length and the affine arc length, respectively. For the

latter, we have Css = na(p, t). Hence, despite the fact that the affine differential geometry is

not defined for non-convex curves, we can define still define the affine curvature evolution

using the Euclidean curvature, by taking the velocity to be k1/3n [AST98].

Related numerical work on motion by mean curvature

The popularity and ubiquity of level set methods has given rise to numerous numerical

methods for the PDE governing the mean curvature evolution (MC) (see the review papers

[DDE05b] and [CMM11] for an extended review).


In the seminal article [OS88], using techniques from the hyperbolic conservation laws,

a scheme is proposed where each level set moves with velocity proportional to the mean

curvature. Crandall and Lions in [CL96] proposed a class of difference schemes for

quasilinear PDE, which include the motion by mean curvature. The schemes, which can be

made monotone by adding a small perturbation as linear diffusion, have some drawbacks,

such as degeneracy and singularities. This makes the scheme complex as several relations

on the parameters must be satisfied in order to ensure its convergence. Finite element

schemes have also been proposed [Wal96, CHR+05, DDE05b], however the theory does

not ensure the uniqueness of solutions.

In [Obe04] a convergent elliptic wide stencil finite difference scheme for ∆1u is pre-

sented based on taking the median of the values of u sampled in a small approximately

circular neighbourhood of x. The motivation follows from observing, using (1.1), that

∆1u = utt, t =(−uy, ux)

(u2x + u2

y)1/2,

where t is the (Euclidean) unit tangent. The median captures an approximation to utt,

the second tangential derivative of u, since the larger values point in the direction of the

gradient and the smaller values point in the opposite directions. Another elliptic scheme

has been proposed by Catté and Dibos [CDK95] which is equivalent to Oberman’s scheme

when the gradient is nonzero, but it lacks consistency otherwise (see [Tak07]).

Numerical schemes have also been proposed for the equivalent geometric curve evolu-

tion. One can interpret it as the singular limit of a semilinear reaction diffusion equation,

which leads to indirect numerical schemes [BG95, ESS92]. Similarly, Bence, Merriman and

Osher [MBO94] proposed a scheme that consists in repeatedly solving the heat equation

for a short time, by using convolution, followed by thresholding. It can be viewed as

well as a singular limit of a reaction diffusion equation. These thresholding methods are

effective for moving a given curve by the evolution, and allow for large time steps to be

taken. Using the equivalent level set representation moves every level set by the evolution,

but requires a much smaller time step.

Related numerical work on motion by affine curvature

Significantly less studied than the mean curvature evolution, there are still some numerical

methods for affine curvature evolution. The recent article [ERT10] gives a Bence-Merriman-

Osher [MBO94] thresholding scheme. It introduced a regularization of the cube root, which

was needed for theoretical purposes, but not in practice.

Alvarez and Guichard proposed a local scheme which lacks the affine invariance

10 1.2. ORGANIZATION OF THE THESIS

property [Gui94]. A morphological scheme which generalized [CDK95] was proposed

by Guichard and Morel for affine curvature [GM97]. This inf-sup scheme, although

morphologically invariant, has some limitations on the speed at which the level set curves

move. In [Moi98], a nonlocal geometric morphological scheme is presented. The author

introduces a geometrical operator, called affine erosion, based on the concept of σ-chords:

region with area σ enclosed by the segment joining two points in the curve and the curve

itself. The affine erosion operator is nonlocal, fully affine invariant and turns out to be

exactly the set of all the middle points of the σ-chords segments of the curve. The author

thus obtains a fast, but difficult to code, algorithm consistent with the curve evolution.

1.2 ORG ANI ZATION OF THE THES IS

The organization of the thesis is as follows: In chapter 2 we review the theory of viscosity

solutions. In chapter 3 we review the theory of monotone, elliptic and filtered schemes.

In chapter 4 we discuss the filtered schemes built for HJ equations, presenting numerical

results in one and two dimensions. In chapter 5, we discuss the numerical methods pro-

posed for the 2-Hessian equation. Computational results are presented on a number of

exact solutions which range in regularity from smooth to nondifferentiable and in shape

from convex to nonconvex. In chapter 6 we build a provably convergent scheme for the

nonlinear PDE that governs the motion of level sets by affine curvature. Numerical experi-

ments are presented to demonstrate the accuracy and stability of the discretization, as well

as the fact that the approximate solutions capture the affine invariance and morphological

properties of the evolution. Finally, in chapter 7 we summarize the results and discuss

future work.

CHAPTER 2

VISCOSITY SOLUTIONS

In this chapter we discuss viscosity solutions, the appropriate notion of solution for

elliptic partial differential equations (PDEs) when working in the framework of Barles and

Souganidis. The standard reference and the one mainly followed here is [CIL92]. It is also

worth mentioning the textbook by Koike [Koi04].

Viscosity solutions were first introduced in a series of papers by Crandall, P. L. Lions

and Evans [Eva80, CL83, CEL84]. The book by Evans [Eva98] also provides a good

introduction to the first order case focusing on Hamilton-Jacobi equations and applications

to control theory.

2.1 MOTIVATION OF VIS C OS ITY S OLUTIONS

In this section, we give a brief example to motivate the use of viscosity solutions in general.

Consider the following Eikonal equation in the one-dimensional setting

|u′(x)| = 1, x ∈ (−1, 1),

u(±1) = 0.

This PDE has no differentiable solution but there are infinitely many solutions that are

differentiable almost everywhere: u(x) = 1 − |x| and u(x) = min1 − |x| , |x| are two

simple examples. However, we are interested in a definition of solution that amongst

other properties satisfies uniqueness. Thus, from the infinite set of almost everywhere

differentiable solutions, we must select a particular one.

Consider then for each ε > 0 the following equation

−εu′′(x) + |u′(x)| = 1, x ∈ (−1, 1),

u(±1) = 0,

where the term εu′′(x) regularizes the equation. Denote its unique differentiable solution

by uε. It is possible to show that

limε→0

uε(x) = 1 − |x| for all x ∈ [−1, 1].

This technique is called vanishing viscosity. As we will see below, this limit captures

11

12 2.2. DEFINITION OF VISCOSITY SOLUTION

exactly the unique viscosity solution of the PDE. The term “viscosity” is inspired by this

method, but in fact has no relation to the actual definition of viscosity solution.

2.2 DEFINITION OF VIS C OS ITY S OLUTION

In this section, we present the definition of viscosity solution. Viscosity solutions can be

defined in (slightly) different ways depending on the context: we can allow for discontinu-

ous viscosity solutions and the boundary conditions can be treated in a viscosity sense or

not. Here, we present the definition of interest for the framework of Barles and Souganidis

[BS91], which we discuss in detail in chapter 3.

We start by considering the PDE

F (x, u(x), ∇u(x), D2u(x)) = 0, x ∈ Ω, (2.1)

where Ω is a bounded open subset of Rn, F ∈ C(R × R × R

n × Sn) and Sn denotes the set of

symmetric n × n matrices. The theory of viscosity solution applies to the PDEs that satisfy

a monotonicity condition.

Definition 2.2.1. The function F : R × R × Rn × Sn → R is degenerate elliptic (in the sense of

[CIL92]) if

F (x, r, p, M) ≤ F (x, r, p, N), for all x ∈ R, r ∈ R, p ∈ Rn, M N,

where N M if dᵀNd ≤ dᵀMd for all d ∈ Rn. If in addition

F (x, r, p, M) ≤ F (x, s, p, M), for all x ∈ R, r ≤ s, p ∈ Rn, M ∈ Sn,

we say that F is proper.

Example 2.1. By this convention, the Laplacian operator is elliptic when written F (M) =

−tr(M). Some authors use the other convention, without the minus sign.

Before presenting the definition of viscosity solutions, we recall the definition of upper

and lower semicontinuous functions as well as the definition of upper and lower semicon-

tinuous envelopes. To be precise, we recall too the definitions of lim sup and lim inf. Given

a function u : U → R, we have

lim supy∈U→x

u(y) := infr>0

sup u(y) | y ∈ U ∩ B(x, r) ,

Chapter 2. Viscosity Solutions 13

and

lim infy∈U→x

u(y) := supr>0

inf u(y) | y ∈ U ∩ B(x, r) ,

where B(x, r) denotes the open ball centred at x with radius r.

Definition 2.2.2. A function u : U → R is upper (resp. lower) semicontinuous at x ∈ U provided

lim supy∈U→x

u(y) ≤ u(x)(

resp. lim infy∈U→x

u(y) ≥ u(x))

.

We denote by USC(U) (resp. LSC(U)) the collection of functions that are upper (resp. lower)

semicontinuous at all points of U .

Definition 2.2.3. The upper (resp. lower) semicontinuous envelopes of a function u : U → R is

defined by

u∗(x) = lim supy∈U→x

u(y)(

resp.u∗(x) = lim infy∈U→x

u(y))

.

We can now give the definition of viscosity solutions.

Definition 2.2.4. Let u denote the function u : Ω → R.

(i) We say that u ∈ USC(Ω) is a viscosity subsolution of (2.1) if for every φ ∈ C2(Ω) such that

u − φ has a local maximum at x ∈ Ω, F (x, u(x), ∇φ(x), D2φ(x)) ≤ 0.

(ii) We say that u ∈ LSC(Ω) is a viscosity supersolution of (2.1) if for every φ ∈ C2(Ω) such

that u − φ has a local minimum at x ∈ Ω, F (x, u(x), ∇φ(x), D2φ(x)) ≥ 0.

(iii) We say that u is a viscosity solution of (2.1) if u is both a viscosity subsolution and a viscosity

supersolution.

Remark 2.1. When checking the definition of a viscosity solution we can limit ourselves to

considering unique, strict, global maxima (minima) of u − φ with a value of zero at the

extremum. Geometrically, this means that φ touches u at x from above (below). See, for

example, [Koi04, Prop 2.2].

Remark 2.2. As it stands, viscosity solutions are continuous. We can allow for discontinuous

viscosity solutions, by saying that a locally bounded function u is a viscosity solution (2.1)

if u∗ is a viscosity subsolution of (2.1) and u∗ is a viscosity supersolution of (2.1).

Despite not including the boundary conditions, the above definition is still of interest

when we discuss comparison principles in the next section.

14 2.2. DEFINITION OF VISCOSITY SOLUTION

In general, we are interested in the boundary value problem (BVP)

F (x, u(x), ∇u(x), D2u(x)) = 0, if x ∈ Ω,

B(x, u(x), ∇u(x)) = 0, if x ∈ ∂Ω.(2.2)

where B ∈ C(∂Ω × R × Rn) is proper.

Definition 2.2.5. Let u denote the function u : Ω → R.

(i) We say that u ∈ USC(Ω) is a viscosity subsolution of (2.2) if for every φ ∈ C2(Ω) such that

u − φ has a local maximum at x ∈ Ω,

F (x, u(x), ∇φ(x), D2φ(x)) ≤ 0, if x ∈ Ω,

min

F (x, u(x), ∇φ(x), D2φ(x)), B(x, u(x), ∇φ(x))

≤ 0, if x ∈ ∂Ω.

(ii) We say that u ∈ LSC(Ω) is a viscosity supersolution of (2.2) if for every φ ∈ C2(Ω) such

that u − φ has a local minimum at x ∈ Ω,

F (x, u(x), ∇φ(x), D2φ(x)) ≥ 0, if x ∈ Ω,

max

F (x, u(x), ∇u(x), D2u), B(x, u(x), ∇φ(x))

≥ 0, if x ∈ ∂Ω.

(iii) We say that u is a viscosity solution of (2.2) if u is both a viscosity subsolution and viscosity

supersolution of (2.2).

Remark 2.3. Alternatively, we can define viscosity solutions in terms of the function

G : Ω × R × Rn × Sn → R given by

G(x, r, p, M) =

F (x, r, p, M), x ∈ Ω,

B(x, r, p), x ∈ ∂Ω.

It is easy to see that

G∗(x, r, p, M) = G∗(x, r, p, M) = F (x, r, p, M), x ∈ Ω,

G∗(x, r, p, M) = minF (x, r, p, M), B(x, r, p), x ∈ ∂Ω,

G∗(x, r, p, M) = maxF (x, r, p, M), B(x, r, p), x ∈ ∂Ω.

Then, u ∈ USC(Ω) is a viscosity subsolution of (2.2) if for every φ ∈ C2(Ω) such that u − φ

has a local maximum at x ∈ Ω, G∗(x, u(x), ∇φ(x), D2φ(x)) ≤ 0. Similarly u ∈ LSC(Ω) is a

viscosity supersolution of (2.2) if for every φ ∈ C2(Ω) such that u − φ has a local minimum

at x ∈ Ω, G∗(x, u(x), ∇φ(x), D2φ(x)) ≥ 0.


Example 2.2. One can check that u(x) = 1 − |x| is the viscosity solution of

|∇u| = 1, x ∈ (−1, 1),

u(−1) = u(1) = 0.

Here we consider F (p) = |p| − 1, which leads to

2.3 COMPARI S ON P RINC IP LE AND UNIQUENES S

In this section, we discuss comparison principles for degenerate elliptic PDEs. In general,

comparison principles tells us that subsolutions lie below supersolutions. The difference

between the different comparison principles presented here lies in the way the behaviour

at the boundary is treated.

We start by discussing what we call a weak comparison principle, where we assume

that the subsolution lies below the supersolution at the boundary. This crucial assumption

makes the result substantially easier to prove.

Definition 2.3.1 (Weak comparison principle). We say that the PDE (2.1) has a weak com-

parison principle if whenever u is a viscosity subsolution and v a viscosity supersolution of (2.1)

in Ω such that u ≤ v on ∂Ω, u ≤ v in Ω.

We start with a result for first order equations. To simplify the presentation, we define

the set of modulus of continuity functions as the set

M =

ω : R+ → R

+ | ω(·) is continuous , limr→0

ω(r) = 0

.

Theorem 2.3.2. Assume that F (x, r, p, M) = H(x, p) − f(x) where

|H(x, p) − H(y, p)| ≤ ωH (|x − y| (1 + |p|)) , for all x, y ∈ Ω, p ∈ Rn,

with ωH ∈ M. Suppose that H has homogeneous degree α > 0 with respect to its second argument,

i.e., there is α > 0 such that

H(x, µp) = µαH(x, p), for all x ∈ Ω, p ∈ Rn, µ > 0.

Suppose as well that f ∈ C(Ω) and that there is σ > 0 such that minx∈Ω

f(x) = σ. Then the PDE

(2.1) satisfies a weak comparison principle.

Proof. See [Koi04] for a proof.

16 2.3. COMPARISON PRINCIPLE AND UNIQUENESS

The general case for second order equations is more subtle and it requires additional

assumptions on the PDE, beyond degenerate ellipticity. We introduce the concept of strictly

proper and structure condition.

Definition 2.3.3. We say that F is strictly proper (in Ω) if there exists γ > 0 such that

γ(s − r) ≤ F (x, s, p, M) − F (x, r, p, M) for all r ≤ s, x ∈ Ω, p ∈ Rn, M ∈ Sn.

Remark 2.4. If F is strictly proper then F is proper.

Definition 2.3.4. We say that F satisfies the weak structure condition if there exists ωF ∈ Msuch that if M, N ∈ Sn and α ∈ R satisfy

−3α

I 0

0 I

≤

M 0

0 −N

≤ 3α

I −I

−I I

,

then

F (y, r, α(x − y), N) − F (x, r, α(x − y), M) ≤ ωF

(

α |x − y|2 + |x − y|)

for all x, y ∈ Ω and r ∈ Rn.

Remark 2.5. If F satisfies the structure condition, then F is degenerate elliptic. Moreover,

the structure condition arises naturally in the proof of the comparison principle. For more

details see [CIL92].

Theorem 2.3.5. Assume that F is degenerate elliptic, strictly proper and satisfies the weak struc-

ture condition. Then the PDE (2.1) satisfies a weak comparison principle.

Remark 2.6. In [IL90], weak comparison principles are obtained for F proper. Instead of

the strictly properness of F , one assumes the existence of δ > 0 such that either u is a

subsolution of

F (x, u(x), ∇u(x), D2u(x)) + δ = 0

or v is a supersolution of

F (x, v(x), ∇v(x), D2v(x)) − δ = 0.

In particular, if F (x, r, p, M) = H(r, p, M) + f(x) is such that there exits α > 0 such that

H(µr, µp, µM) = µαH(r, p, M), for all µ > 0, r ∈ R, p ∈ Rn, M ∈ Sn,

and there exists σ > 0 such that minx∈Ω

f(x) = σ then a subsolution (resp. supersolution) of

(2.1) can be perturbed to produce a strict subsolution (resp. supersolution) as above. Thus,


in such case, F satisfies a weak comparison principle. This includes the Monge-Ampère

equation and k-Hessian equations.

The weak comparison principle discussed above is a useful result. For instance, it

implies uniqueness.

Proposition 2.3.6. Assume that the PDE (2.1) has a weak comparison principle. Let u and v be

viscosity solutions of (2.1) in Ω with u = v on ∂Ω. Then u = v in Ω.

Proof. Since u (resp. v) and v (resp. u) are viscosity subsolution and supersolution, re-

spectively, and since u ≤ v (resp. v ≤ u) on ∂Ω, the weak comparison principle yields

u ≤ v (resp. v ≤ u) in Ω. Combining the two inequalities together, we get u = v in Ω as

desired.

However, when building numerical schemes for degenerate elliptic PDEs, in order

to prove their convergence using the Barles-Souganidis theorem, one needs a strong

comparison principle where the boundary conditions are treated in the viscosity sense.

Definition 2.3.7 (Strong comparison principle). We say that the BVP (2.2) has a strong

comparison principle if whenever u is a viscosity subsolution and v a viscosity supersolution of

(2.2), u ≤ v on Ω.

Definition 2.3.8. We say that F satisfies the strong structure condition if there exists ωF ∈ Msuch that if M, N ∈ Sn and α ∈ R satisfy

−3α

I 0

0 I

≤

M 0

0 −N

≤ 3α

I −I

−I I

,

then

F (y, r, p, N) − F (x, r, p, X) ≤ ωF

(

α |x − y|2 + |x − y| (|p| + 1))

for all x, y ∈ Ω, r ∈ Rn and p ∈ R

n.

Theorem 2.3.9. Let B : ∂Ω × ×R × Rn be given by B(x, r, p) = 〈n(x), p〉 + f(x, r), where

f ∈ C(∂Ω × R) is nondecreasing in r for each x ∈ ∂Ω and n(x) denotes the outward unit normal

to x ∈ ∂Ω. Assume that Ω is a compact C1 n-submanifold with boundary of Rn that satisfies the

uniform exterior sphere condition: there exists r > 0 such that B(x + rn(x), r) ∩ Ω = ∅ for all

x ∈ ∂Ω. Assume as well that there is a neighbourhood V of ∂Ω and ωV ∈ M such that

|F (x, r, p, M) − F (x, r, q, N)| ≤ ωV (|p − q| + ‖M − N‖)

18 2.3. COMPARISON PRINCIPLE AND UNIQUENESS

for all x ∈ V , p, q ∈ Rn, M, N ∈ Sn. Suppose as well that F is degenerate elliptic, strictly proper

in Ω and satisfies the strong structure condition. Then the BVP (2.2) satisfies a strong comparison

principle.

The above theorem deals with the case of Neumann boundary conditions. The case

of Dirichlet boundary conditions (B(x, r, p) = r − g(x)) is however more subtle. If the

viscosity subsolution and viscosity supersolution are assumed to be continuous at the

boundary, then the strong comparison principle holds (see [CIL92] for details).

We are interested in the application of the strong comparison principle in the Barles

and Souganidis theorem. In such a framework and in some special cases, it is possible to

show that the boundary conditions are satisfied pointwise by the viscosity subsolution

u and viscosity supersolution v (i.e. u(x) = v(x) = g(x) for all x ∈ ∂Ω), in which case the

strong comparison principle follows from the weak comparison principle. This was the

case in [FJ16].

A strong comparison principle is not available in general in the Dirichlet case when

there exists a viscosity solution that does not satisfy the boundary conditions pointwise.

This is illustrated for the eikonal equation in Example 2.3 below. Another example is

provided in [Fro16a] for the prescribed Gauss curvature problem.

Example 2.3. Consider the function

u(x) =

x, x ∈ (0, 1),

2, x = 1.

One can show that u is a viscosity subsolution of

|u′(x)| = 1, x ∈ (0, 1),

u(0) = 0,

u(1) = 2.

One the other hand, the function u(x) = x is a viscosity solution. This simple example

shows that the Eikonal equation does not satisfy a strong comparison principle: u is a

viscosity subsolution that does not lie below the viscosity supersolution u. However, it is

true that u ≤ u in Ω. This was precisely the result proven in [Fro16b] for the prescribed

Gauss curvature problem: viscosity subsolutions lie below viscosity supersolutions in Ω.

Notice that the viscosity solution u does not satisfy the boundary conditions in the strong

sense.


Proving a strong comparison principle, in particular in the case of the Dirichlet bound-

ary conditions, is still the subject of current research and not the focus of this thesis.

Moreover, we are interested in problems with continuous viscosity solutions. Thus, we

will assume that the problems considered here satisfy a strong comparison principle.

2.4 EXI S TENC E OF VIS C OS ITY S OLUTIONS

In this section, we briefly discuss the existence of viscosity solutions.

In general, existence of viscosity solutions follows from Perron’s method, although in

some specific cases there exists solution formulas. This is the case of Bellman and Isaacs

equations (see [Koi04]) or the Hopf-Lax formula for some Hamilton-Jacaboi equations (see

[Eva98]).

Similarly to the results obtained for the strong comparison principle, we have slightly

different results depending on the type of boundary conditions, although both results

follow from Perron’s method.

For the Neumman problem, existence follows under the same assumptions used to

prove a strong comparison principle.

Theorem 2.4.1. Under the assumptions of Theorem 2.3.9, there exists a viscosity solution (2.2).

As for the Dirichlet problem, we have the following result.

Theorem 2.4.2. Let B(x, r, p) = r − g(x) and suppose that (2.1) has a weak comparison principle.

Suppose also that there is a strong viscosity subsolution u and a strong viscosity supersolution u of

(2.1) that satisfy the boundary condition u∗(x) = u∗(x) = g(x) for x ∈ ∂Ω. Then

W (x) = sup

w(x) | u ≤ w ≤ u in Ω and w is a strong viscosity subsolution of (2.1)

is a viscosity solution of (2.1) with W (x) = g(x) for all x ∈ ∂Ω.

Remark 2.7. Although the existence of viscosity subsolution and viscosity supersolution

(that satisfy the boundary conditions) is not present in Theorem 2.4.1, their existence

follows from the assumptions in the theorem.

CHAPTER 3

NUMERICAL SCHEMES

In this chapter, we discuss the framework developed in [BS91] that provides conditions

under which approximation schemes converge to the unique viscosity solution of an

elliptic PDE. We will focus on finite difference schemes, but the notions of monotone,

elliptic and filtered schemes can be extended to different frameworks (e.g. finite elements).

3.1 MONOTONE FINI TE DIFFER ENCE S C HEMES

Let D be the computational domain such that Ω ⊆ D and consider a set of discretization

points Gh ⊆ D. Here h is a small parameter relating to the grid resolution, which we

assume to be such that

limh→0

supy∈Ω

minx∈Gh

|x − y| = 0. (3.1)

Let C(Gh) denote the set of grid functions u : Gh → R. Define

GhV = Ω ∩ Gh, ∂Gh = Gh \ Gh

V .

For simplicity, we assume that ∂Gh ⊆ ∂Ω.

Example 3.1. Typically, we take Ω = D to be a n-cube and discretize with a uniform

grid spacing h. For instance in the one dimensional setting (n = 1), if D = [0, 1], then

Gh = x ∈ hZ | x ∈ D and ∂Gh = 0, 1.

Definition 3.1.1. A finite difference scheme is a map F h : C(Gh) → C(Gh), such that given

u ∈ C(Gh) we write, for x ∈ Gh,

F h[u](x) = F h(x, u(x), u(·)), (3.2)

where u(·) indicates the values of the grid function u. We say the finite difference scheme F h has a

stencil of width W if F h depends only on values u(y) for ‖y − x‖∞ ≤ Wh. A solution of the finite

difference scheme F h is a grid function u ∈ C(Gh) which satisfies the equation

F h[u](x) = 0 for all x ∈ Gh. (3.3)

Remark 3.1. The explicit dependence on u(x), which we refer to as the reference variable,

will become clear later when we define elliptic finite difference schemes.

21

22 3.1. MONOTONE FINITE DIFFERENCE SCHEMES

Definition 3.1.2. The finite difference scheme (3.2) is monotone if as function F h : Gh × R ×C(Gh) → R we have

u(·) ≥ v(·) =⇒ F h(x, r, u(·)) ≤ F h(x, r, v(·))

for all x ∈ Gh, r ∈ R and u, v ∈ C(Gh).

Definition 3.1.3. The finite difference scheme F h (3.2) is consistent with the BVP (2.2) (in the

sense of [BS91]) if for any smooth function φ and x ∈ Ω,

lim suph→0,y∈Gh→x,ξ→0

F h(y, φ(y) + ξ, φ(·) + ξ) ≤ G∗(x, u(x), ∇φ(x), D2φ(x)),

lim infh→0,y∈Gh→x,ξ→0

F h(y, φ(y) + ξ, φ(·) + ξ) ≥ G∗(x, u(x), ∇φ(x), D2φ(x)).

where

G(x, r, p, M) =

F (x, r, p, M), x ∈ Ω,

B(x, r, p), x ∈ ∂Ω.

Remark 3.2. Most schemes will have only h as a parameter. For those, we say the scheme is

accurate to order k if for any smooth function φ and x ∈ Gh ∩ Ω

F h[φ](x) = F (x, φ(x), ∇φ(x), D2φ(x)) + O(hk).

The order of accuracy of a scheme (and its consistency) are usually verified by a Taylor

series argument.

Stability is a desirable property in numerical methods, but the precise definition de-

pends on the context. Here we follow [BS91].

Definition 3.1.4. The finite difference scheme F h (3.2) is stable if there exists h0 > 0 such that

for all 0 < h < h0 any solution u ∈ C(Gh) of (3.3) is bounded independently of h.

Our finite difference schemes and their solutions are defined only on a finite set of

discretization points Gh and not on the entire domain Ω as in [BS91]. Therefore, we must

modify the original proof of Barles and Souganidis to account for the different framework.

This is accomplished here with a nearest neighbour extension: letting Uh ∈ C(Gh) be a

solution the approximation scheme on the grid, we define the piecewise constant extension

as

uh(x) = max

Uh(y) | y ∈ Gh, |y − x| = minz∈Gh

|z − x|

(3.4)

Chapter 3. Numerical Schemes 23

for all x ∈ Ω. Likewise, the finite difference schemes are also extended to functions defined

on the whole domain Ω:

F h[u](x) = max

F h[u](y) | y ∈ Gh, |y − x| = minz∈Gh

|z − x|

(3.5)

for all x ∈ Ω.

Remark 3.3. The extension of F h (3.5) is still monotone as a function F h : Ω ×R× C(Ω) → R.

Indeed, let x ∈ Ω, r ∈ R and u, v ∈ C(Ω) with u(·) ≥ v(·). Then

F h(x, r, u(·)) = F h(y, r, u(·)) for some y ∈ Gh,

≤ F h(y, r, v(·)) since F h is monotone and u(·) ≥ v(·),≤ F h(x, r, v(·)) by (3.5).

We are then ready to present the Barles and Souganidis theorem. The proof is similar

to the proof of Theorem 3.4.3.

Theorem 3.1.5. Assume the BVP (2.2) satisfies a strong comparison principle (see Definition

2.3.7). Let F h by any stable, monotone finite difference scheme consistent with the BVP (2.2) and

Uh ∈ C(Gh) any solution of (3.3). Then

limh→0

uh(x) = u(x), for all x ∈ Ω,

where uh is the piecewise constant extension of Uh (3.4) and u is the viscosity solution of (2.2).

3.2 ELLIP TI C FINITE DI FFERENC E S C HEMES

The definition of monotone schemes does not provide any insight on how to build them.

In this section, we discuss elliptic schemes, introduced by Oberman in [Obe06a]. These

not only guarantee monotonicity but are also easy to build as they draw inspiration from

finite difference approximations.

Definition 3.2.1. A finite difference scheme F h is elliptic if it can be written as

F h[u](x) = F h(x, u(x), u(x) − u(·)),

where F h is nondecreasing in its second and third arguments, i.e.,

r ≤ s, u(·) ≤ v(·) =⇒ F h (x, r, u(·)) ≤ F h(x, s, v(·)),

for all x ∈ Gh, r, s ∈ R and u, v ∈ C(Gh).

24 3.2. ELLIPTIC FINITE DIFFERENCE SCHEMES

Example 3.2. The backward approximation

u(x) − u(x − h)

h= ux(x) + O(h)

is an elliptic scheme for ux. Similarly, the forward approximation

−u(x + h) − u(x)

h= −ux(x) + O(h)

is an elliptic scheme for −ux.

Proposition 3.2.2. Elliptic finite difference scheme are monotone.

Proof. Let x ∈ Gh, r ∈ R and u, v ∈ C(Gh). Then

u(·) ≥ v(·) =⇒ r − u(·) ≤ r − v(·) =⇒ F h (x, r, r − u(·)) ≤ F h(x, r, r − v(·))

since F h is elliptic.

As the name suggests, elliptic finite difference schemes are the discrete counterpart of

degenerate elliptic operators. Ideally, we would like that these approximation schemes

inherit the basic structure of the underlying degenerate elliptic PDE. In particular, we

note that under similar assumptions, the elliptic finite difference schemes enjoy a discrete

comparison principle.

Definition 3.2.3. The comparison principle holds for the finite difference operator F h : C(Gh) →C(Gh), if F h[u] ≤ F h[v] implies u ≤ v, more precisely,

F h[u](x) ≤ F h[v](x) for all x ∈ Gh =⇒ u(x) ≤ v(x) for all x ∈ Gh. (3.6)

Remark 3.4. In the discrete comparison principle (3.6), the boundary conditions are encoded

in F h. For Dirichlet boundary conditions, we define

F h[u](x) = u(x) − g(x), for all x ∈ ∂Gh.

Thus, the assumption F h[u] ≤ F h[v] means u ≤ v at boundary points. Uniqueness of

solutions follows from the discrete comparison principle, since if u, v are solutions, then

F h[u] = F h[v] = 0, so u ≤ v and u ≥ v, and thus u = v.

Definition 3.2.4. The finite difference scheme F h is proper if is strictly increasing on its second

argument, i.e., there exists δ > 0 such that

r ≤ s =⇒ F h(x, r, u(·)) − F h(x, s, u(·)) ≤ δ(r − s)


for all x ∈ Gh and u ∈ C(Gh).

Remark 3.5. Without any loss of generality, we can assume the scheme to be proper. Indeed,

if the scheme is not proper, we can consider instead the scheme F h[u] + εu for arbitrarily

small ε (for example, smaller than the discretization error).

Theorem 3.2.5. If F h is a proper elliptic finite difference scheme, then a comparison principle (3.6)

holds for F h.

Proof. See [Obe06a].

The finite difference schemes may be nonlinear and nondifferentiable, but we will

require them to be Lipschitz continuous.

Definition 3.2.6. The finite difference scheme F h : C(Gh) → C(Gh) is Lipschitz continuous with

constant Kh if Kh is the smallest constant such that

∣

∣

∣F h (x, r, u(·)) − F h (x, s, v(·))∣

∣

∣ ≤ Kh max(|r − s|, ‖u − v‖∞), for all x ∈ Gh.

Remark 3.6. In the definition above, the maximum on the right hand side can be replaced

with a maximum over the neighbouring grid values without changing the Lipschitz

constant.

As we saw in the previous section, we require that our schemes are well-posed in the

sense that solutions exist and are stable. Here we will present a result that shows that these

properties follow from the existence of strict classical subsolutions and supersolutions,

whose definition we give below.

Definition 3.2.7. A function u ∈ C2(Ω) is a strict classical subsolution of (2.2) if there exists

some µ > 0 such that

F (x, u(x), ∇u(x), D2u(x)) ≤ −µ, if x ∈ Ω,

min

F (x, u(x), ∇u(x), D2u), B(x, u(x), ∇u(x))

≤ −µ, if x ∈ ∂Ω.

A function u ∈ C2(Ω) is a strict classical supersolution of (2.2) if there exists some µ > 0 such that

F (x, u(x), ∇u(x), D2u(x)) ≥ µ, if x ∈ Ω,

max

F (x, u(x), ∇u(x), D2u), B(x, u(x), ∇u(x))

≥ µ, if x ∈ ∂Ω.

Lemma 3.2.8. Let F h be elliptic Lipchitz finite difference scheme consistent with the BVP (2.2).

Suppose also that there exists functions v, w ∈ C2(Ω) such that v is a strict subsolution and w is a

26 3.2. ELLIPTIC FINITE DIFFERENCE SCHEMES

strict supersolution of (2.2). Then for sufficiently small h > 0, the approximation scheme F h has a

solution and F h is stable.

Proof. See [Fro16b].

Remark 3.7. Alternatively, in [Obe06a] the author shows that if a scheme is proper, Lipschitz

continuous and elliptic, then solutions exist and are stable.

Although the framework is defined in the context of stationary equation, it can be used

to treat time dependent parabolic equations of the form ut + F [u] = 0. Below we give

sufficient conditions to build a monotone scheme for these equations. The time derivative

in the equation is treated with a forward Euler step which leads to definition of the explicit

Euler map.

Definition 3.2.9. For ρ > 0, define the Euler map Sρ : C(Gh) → C(Gh) by

Sρ[u] = u − ρF h[u].

Proposition 3.2.10. Let F h be a Lipschitz continuous, elliptic scheme with Lipschitz constant Kh.

Assume that the CFL condition ρ ≤ 1/Kh is satisfied. Then

u(·) ≤ v(·) =⇒ Sρ[u] ≤ Sρ[v]

for all u, v ∈ C(Gh).

Proof. See [Obe06a].

Remark 3.8. In addition, if we assume that F h is proper and the strict inequality ρ < 1/Kh

holds, F h has a unique solution and the iterates of the Euler map converge to the solution

for arbitrary initial data. In other words, the Euler map Sρ is a convergent solver to the

unique solution of F h[u] = 0.

The approximate solutions uh,dt of ut + F [u] = 0 with initial condition u(x, 0) = u0(x)

are then defined as

uh,dt(x, n + 1) = Sdt[u(·, n)],

uh,dt(x, 0) = u0(x)

for all x ∈ GhV , n = 0, 1, . . ., with the appropriate boundary conditions. The underlying

finite difference scheme is monotone, a direct consequence of Proposition 3.2.10 provided

dt satisfies the CFL condition. In this specific context of parabolic equations, monotonicity


means that the following discrete comparison principle holds: if u(x, n), v(x, n) are solu-

tions of the scheme, u(x, n) ≤ v(x, n) for all x ∈ Gh implies u(x, n + 1) ≤ v(x, n + 1) for all

x ∈ Gh.

3.3 BUILDING ELLIP TI C SC HEMES

In order to build elliptic finite difference schemes, we study the properties of nonde-

creasing maps. This is required since in general elliptic schemes are built composing

nondecreasing maps with elliptic terms. Moreover, for some nonlinear elliptic PDEs the

domain of ellipticity is restricted and thus we need to build nondecreasing extensions

of the underlying functions. This is the particular case of the 2-Hessian equation (see

chapter 5). In this section we define elliptic schemes for |∇u| and − |∇u|. This is done both

as an illustration of the general principle, and because these schemes will be used later in

the thesis.

We start with some definitions, that also serve as examples.

Definition 3.3.1. For u : R2 → R, define the standard finite differences

uhx(x, y) :=

u(x + h, y) − u(x − h, y)

2h,

uhxx(x, y) :=

u(x + h, y) − 2u(x, y) + u(x − h, y)

h2,

uhxy(x, y) :=

u(x + h, y + h) + u(x − h, y − h) − u(x + h, y − h) − u(x − h, y + h)

4h2,

(3.7)

for ux, uxx, uxy, respectively, and, similarly for uhy , and uh

yy in the y coordinate. These are second

order accurate approximations. Only the operator −uhxx is elliptic, the others are not.

Definition 3.3.2. Define the backward and forward first order derivatives

D−x [u](x, y) :=

u(x, y) − u(x − h, y)

h= ux(x, y) + O,

−D+x [u](x, y) :=

u(x, y) − u(x + h, y)

h= −ux(x, y) + O,

and, similarly D−y [u] and D+

y [u]. Both D−x [u] and −D+

x [u] are elliptic.

Definition 3.3.3 (nondecreasing functions). For x, y ∈ RN we say x ≤ y if xi ≤ yi for all

i = 1, . . . , N . We say the function F : RN → R is nondecreasing, and write F ∈ ND(RN), if

x ≤ y =⇒ F (x) ≤ F (y).

28 3.3. BUILDING ELLIPTIC SCHEMES

Write R+ = x ∈ R | x ≥ 0 and R

− = x ∈ R | x ≤ 0. Furthermore, if F ∈ ND(RN) and

F : RN → R

+, (resp. F : RN → R

−) we write F ∈ ND+(RN) (resp. F ∈ ND−(RN)).

Remark 3.9. When f ∈ C1(RN), if f is nondecreasing in each variable, i.e., fxi≥ 0 for all

i = 1, . . . , N , then f ∈ ND(RN).

Remark 3.10. The set of nondecreasing functions is closed under the composition of func-

tions. In particular, sums of nondecreasing functions are nondecreasing.

We give some simple examples in the form of definitions as they are important on their

on: they constitute the building blocks of some of the elliptic schemes proposed in this

thesis.

Definition 3.3.4. The function x+ = max(x, 0) ∈ ND+(R) and x− = min(x, 0) is in ND−(R).

Definition 3.3.5. Write N(x, y) =√

x2 + y2, and define

N+(x, y) := N(x+, y+) and N−(x, y) := −N(x−, y−).

Then N+ ∈ ND+(R2), and N− ∈ ND−(R2). Furthermore, N+ = N on x, y ≥ 0, N− = −N

on x, y ≤ 0.

Example 3.3 (Upwinding). More generally, if we consider the operator a(x)ux, then the

upwind discretization

a+D−x [u] + a−D+

x [u]

is first order accurate and elliptic.

We now present two simple examples of composing nondecreasing maps with elliptic

terms to build elliptic schemes.

Example 3.4. Define

∣

∣

∣uhx

∣

∣

∣

+= max

−D+x [u], D−

x [u], 0

, −∣

∣

∣uhx

∣

∣

∣

−= min

−D+x [u], D−

x [u], 0

that approximate |ux| and −|ux| to first order. The operators∣

∣

∣uhx

∣

∣

∣

+and −

∣

∣

∣uhx

∣

∣

∣

−are elliptic,

the former is nonnegative, and the latter is nonpositive.

Proof. Since D−x and −D+

x are elliptic and max, min ∈ ND(R2), the composed functions∣

∣

∣uhx

∣

∣

∣

+and −

∣

∣

∣uhx

∣

∣

∣

−are elliptic.

Example 3.5. Define

|∇uh|+ = N(

|uhx|+, |uh

y |+)

, −|∇uh|− = −N(

−|uhx|−, −|uh

y |−)


which are elliptic, consistent with |∇u|, and − |∇u|, respectively, and first order accurate.

Proof. Since |uhx|+ and |uh

y |+ are nonnegative,

|∇uh|+ = N(

|uhx|+, |uh

y |+)

= N+(

|uhx|+, |uh

y |+)

.

Thus, since N+ ∈ ND(R2) and |uhx|+ and |uh

y |+ are elliptic, the composed function |∇uh|+is elliptic.

Similarly, −|uhx|− and −|uh

y |− are elliptic and nonpositive and

−|∇uh|− = N−(

−|uhx|−, −|uh

y |−)

with N− ∈ ND(R2). Thus −|∇uh|− is elliptic.

Consistency and first order accuracy follow from the generalized chain rule: the

discretization of each term is consistent and first order accurate, and N(·) is a 1-Lipschitz

function.

3.4 F ILTERED FI NITE DIFFER ENC E S CHEMES

Although monotone schemes are provably convergent, they are only first (resp. second)

order accurate for first (resp. second) order equations [Obe06b]. The filtered schemes

discussed here are built to achieve higher accuracy while retaining the convergence prop-

erty of the monotone schemes. We start the section with motivating example for filtered

schemes. Then we introduce nearly monotone schemes, a general class of approximation

schemes that includes the filtered schemes, which are introduced after. We choose to focus

on nearly monotone schemes since they capture the underlying reason why the original

result of Barles and Souganidis [BS91] can be exploited, while also providing a general

framework to define filtered schemes in different ways. Indeed, this is done in chapter 6

when building filtered schemes for the PDE that governs the motion of level sets by affine

curvature.

Example 3.6. Consider the one-dimensional Eikonal

|u′(x)| = 1, x ∈ (−1, 1),

u(±1) = 0.

The finite difference schemes

∣

∣

∣uhx

∣

∣

∣

M:= max

−D+x [u], D−

x [u], 0

and∣

∣

∣uhx

∣

∣

∣

A:=

|u(x + h) − u(x − h)|2h

30 3.4. FILTERED FINITE DIFFERENCE SCHEMES

are monotone and accurate, respectively. These are, respectively, the schemes∣

∣

∣uhx

∣

∣

∣

+and

∣

∣

∣uhx

∣

∣

∣ introduced above. The different notation used in this example is to better illustrate

the general idea behind the filtered schemes. Theorem 3.1.5 guarantees the convergence

of the solutions of the monotone scheme to the unique viscosity solution of the PDE. The

accurate scheme is unstable, in particular when used to compute singular solutions, which

suggests we should only use it in regions where the solution is smooth.

Notice that, up to a scaling in h, the difference of the schemes is the centered finite

difference approximation for |uxx|:∣

∣

∣

∣

∣

∣

∣uhx

∣

∣

∣

A −∣

∣

∣uhx

∣

∣

∣

M∣

∣

∣

∣

=h

2

|u(x + h) − 2u(x) + u(x − h)|h2

=h

2

∣

∣

∣uhxx(x)

∣

∣

∣ .

We can then use this as a (local) criteria to decide whether or not to use the accurate scheme.

This leads us to consider the following (filtered) scheme

∣

∣

∣uhx

∣

∣

∣

F=

∣

∣

∣uhx

∣

∣

∣

A, if

∣

∣

∣

∣

∣

∣

∣uhx

∣

∣

∣

A −∣

∣

∣uhx

∣

∣

∣

M∣

∣

∣

∣

≤√

h,

∣

∣

∣uhx

∣

∣

∣

M, otherwise.

(The choice of the factor√

h will be addressed later.) The main idea here is that we decide

which scheme to use by looking into the size of the difference between the monotone

scheme and the accurate scheme, instead of an approximate smoothness criteria. By doing

so, we have∣

∣

∣uhx

∣

∣

∣

F=∣

∣

∣uhx

∣

∣

∣

M+ O(

√h),

which is the crucial property for the convergence proof as we will see below.

We now introduce nearly monotone finite difference schemes

Definition 3.4.1. The finite difference scheme F hN is a perturbation if there is w ∈ M such that

supu∈C(Gh)

supx∈Gh

∣

∣

∣F hN [u](x)

∣

∣

∣ ≤ w(h).

Definition 3.4.2 (Nearly monotone finite difference scheme). A finite difference scheme F h is

nearly monotone if it can be written as

F h[u] = F hM [u] + F h

N [u],

where F hM is a monotone scheme and F h

N is a perturbation.

Nearly monotone schemes are still provably convergent. The convergence proof follows

from a simple adaptation of the Barles and Souganidis convergence proof [BS91]: the


small (uniformly bounded) correction to the scheme due to the lack of monotonicity can

be absorbed into the term usually seen as the consistency error. In fact, in [BS91] the

convergence of “almost monotone” schemes is mentioned as a remark, but no definition

or examples are given.

Theorem 3.4.3. Assume the BVP (2.2) satisfies a strong comparison principle (see Definition

2.3.7). Let F h be any stable, nearly monotone finite difference scheme consistent with the BVP (2.2)

and Uh ∈ C(Gh) any solution of (3.3). Then

limh→0

uh(x) = u(x), for all x ∈ Ω,


Proof. We follow the proof in [FO13], which itself is a modification of the original proof

in [BS91] to include the case of nearly monotone schemes. Here, following [Fro16a], a

simple adaptation is done to account for the fact that solutions of (3.3) are only defined on

a grid. This is achieved by extending the solutions to the entire domain using the nearest

neighbour extension (3.4) as well as the finite difference schemes (3.5).

Define

u(x) = lim suph→0,y→x

uh(y) ∈ USC(Ω) and u(x) = lim infh→0,y→x

uh(y) ∈ LSC(Ω)

From the stability of the solutions Uh, it follows that both u and u are bounded. In addition,

we know that u ≤ u by the definition on lim sup and lim inf.

Assume for now that u is a viscosity subsolution and u is a viscosity supersolution.

Then from the strong comparison principle for (2.2) applied to u and u, we conclude that

u ≤ u. We can then conclude that u := u = u and therefore u is the unique solution of

(2.2), again by the comparison principle for (2.2). The convergence then follows from the

definitions of u and u.

It then remains to show the claim that u is a viscosity subsolution and u is a viscosity

supersolution. We proceed to show that u is a viscosity subsolution since the proof for u is

similar.

Given a smooth test function φ, let x0 ∈ Ω be a strict global maximum of u with

φ(x0) = u(x0). By Lemma 3.4.4 below, we can find sequences with

hn → 0

yn → x0

uhn(yn) → u(x0)


where yn is a global maximizer of uhn − φ.

Define

εn = uhn(yn) − φ(yn). (3.8)

Then εn → u(x0) − φ(x0) = 0 and uhn(x) − φ(x) ≤ uhn(yn) − φ(yn) = εn for any x ∈ Ω. In

particular,

uhn(·) ≤ φ(·) + εn. (3.9)

We know that

u(·) ≥ v(·) ⇒ F hM(x, r, u(·)) ≤ F h

M(x, r, v(·))

for all x ∈ Gh, r ∈ R and u, v ∈ C(Gh) due to the monotonicity of the scheme (Definition

3.1.2). Using now the fact that F h is nearly monotone (Definition 3.4.2) we get that

u(·) ≥ v(·) ⇒ F h(x, r, u(·)) − 2w(h) ≤ F h(x, r, v(·)),

for all x ∈ Gh, r, s ∈ R and u, v ∈ C(Gh). Hence from (3.9) we conclude that

F hn(x, r, φ(·) + εn) − 2w(hn) ≤ F hn(x, r, uhn(·)) (3.10)

for all x ∈ Gh and r ∈ R. We then have

0 = F hn [uhn ](yn) since uhn is a solution

= F hn(yn, uhn(yn), uhn(·))= F hn(yn, φ(yn) + εn, uhn(·)) by (3.8)

≥ F hn(yn, φ(yn) + εn, φ(·) + εn) − 2w(hn) by (3.10).

Finally, taking the lim inf we get

0 ≥ lim infn→∞

F hn(yn, φ(yn) + εn, φ(·) + εn) − 2w(hn)

≥ lim infhn→0,y→x0,ε→0

F hn(y, φ(y) + ε, φ(·) + ε)

= F∗(x0, φ(x0), ∇φ(x0))

= F∗(x0, u(x0), ∇φ(x0))

which shows that u is a subsolution.

In the above proof we required the use of the following Lemma whose proof we present

now for completion.


Lemma 3.4.4. Suppose the family of function uh is bounded uniformly in h. Define

u(x) = lim suph→0,y→x

uh(u) ∈ USC(Ω).

Given a smooth function φ, let x0 be a strict global maximum of u − φ with u(x0) = φ(x0). Then

there are sequences

hn → 0

yn → x0

uhn(yn) → u(x0)

where yn is a global maximizer of uhn − φ.

Proof. From the definition of lim sup, there are sequences such that

hn → 0,

zn → x0,

uhn(zn) → u(x0).

Let yn ∈ Ω be the global maximizers of uhn(·) − φ(·). Then we have

uhn(yn) − φ(yn) ≥ uhn(zn) − φ(zn) → u(x0) − φ(x0) = 0.

In addition, for any δ > 0 and large enough n,

uhn(yn) − φ(yn) ≤ u(yn) − φ(yn) + δ ≤ u(x0) − φ(x0) + δ = δ

where we used the fact that x0 is a global maximum of u − φ with u(x0) = φ(x0). Thus we

conclude that

uhn(yn) − φ(yn) → 0.

Now, we show by contradiction that yn → x0. Suppose not. Then, by passing to a

subsequence if needed there is an R > 0 such that |yn − x0| > R. Moreover, since u − φ has

a strict, global and unique maximum at x0 with value zero, there is a K > 0 such that

u(y) − φ(y) < −K

whenever |y − x0| > R. For n large enough we have

uhn(yn) ≤ u(yn) +K

2


and so

uhn(yn) − φ(yn) ≤ u(yn) − φ(yn) +K

2< −K +

K

2= −K

2

which contradicts the fact that uhn(yn) − φ(yn) → 0. We then conclude that yn → x0.

Finally we see that

∣

∣

∣uhn(yn) − u(x0)∣

∣

∣ =∣

∣

∣uhn(yn) − φ(x0)∣

∣

∣

≤∣

∣

∣uhn(yn) − φ(yn)∣

∣

∣+ |φ(yn) − φ(x0)|→ 0

and therefore uhn(yn) → u(x0) as desired.

The stability of nearly monotone finite difference schemes is one of the assumptions in

Theorem 3.4.3. The next result addresses precisely that. It tells us that not only solutions

for the nearly monotone finite difference scheme exist but that also that these are stable,

provided the perturbation is continuous and the well-posedness of the underlying mono-

tone schemes. The latter assumption was shown to be true in [Obe06a] for finite difference

schemes that are elliptic, proper and Lipschitz continuous.

Proposition 3.4.5. Let S denote the continuous solution operator for the inhomogeneous problem

for the monotone scheme

F hM [u] + g = 0,

for which we suppose that solutions exist. Suppose as well that there exists h0 > 0 such that for all

0 < h < h0 and g ∈ C(Gh) any u ∈ C(Gh) such that F hM [u] + g = 0 is bounded independently of

h. Let F hN be a continuous perturbation and denote by F h[u] = F h

M [u]+F hN [u] the nearly monotone

scheme. Then solutions of F h[u] = 0 exist and F h is stable.

Remark 3.11. Notice that we do not prove that the solutions of the nearly monotone scheme

are unique: only stability is required to to apply Theorem 3.4.3. Moreover, we emphasize

that this result can only be applied to the filtered schemes defined below, when the filter

function is continuous.

Proof. We follow [FO13].

F hN is a perturbation. Therefore there exists C > 0, independent of h, such that

∥

∥

∥F hN [u]

∥

∥

∥ ≤ C

for all u ∈ C(Gh). By assumption S is continuous and so

∥

∥

∥S(F hN)∥

∥

∥ ≤ R


for some R > 0, also independent of h. Since u ∈ C(Gh) is a solution of F h[u] = 0 if and

only if it is a fixed point of S F hN , the above inequality allows us to conclude that F h is

stable. Moreover, it follows that

S(F hN [BR]) ⊆ BR,

where here BR denotes the ball u ∈ C(Gh) : ‖u‖ ≤ R. Hence, by the Brouwer’s fixed

point theorem and due to the continuity of S F hN , we conclude that there exists a fixed

point u∗ of S F hN , i.e., u∗ = S(F h

N [u∗]) which means that

F hM [u∗] + F h

N [u∗] = 0.

Thus u∗ is a solution of F h[u] = 0.

The filtered schemes we define here fit very naturally into the framework of nearly

monotone schemes, while also being general enough to allow for a variety of schemes. The

main idea is to provide a systematic method to blend a monotone scheme with an accurate

scheme and retain the advantages of both: convergence of the former and higher accuracy

of the latter. This is achieved by requiring that the difference between the filtered scheme

and the monotone scheme is uniformly bounded, in other words, that the schemes are

nearly monotone. Filtered schemes were introduced in [FO13] in the context of the Monge-

Ampère equation. There they were used to overcome the reduction in accuracy based

on the use of a wide-stencil monotone scheme that introduces a directional resolution

error. The idea of blending a monotone scheme with an accurate scheme was first seen in

[Abg09] in the context of Hamilton-Jacobi equations.

Definition 3.4.6 (Filter function). We say that S : R → R is a filter function if it is a bounded

function that is equal to the identity in a neighbourhood of the origin and zero away from the origin.

Example 3.7. The following functions, depicted in Figure 3.1,

S(x) =

x, |x| ≤ 1,

0, |x| > 1,and S(x) =

x, |x| ≤ 1,

0, |x| ≥ 2,

−x + 2, 1 ≤ x ≤ 2,

−x − 2, −2 ≤ x ≤ −1.

are examples of filter functions. The former is discontinuous, while the latter is continuous.


-2 -1 1 2

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

Figure 3.1: Examples of filter functions: discontinuous filter (left), continuous filter (right).

Definition 3.4.7. Let S be a filter function and ε(h) : R+ → R

+ be a nonnegative modulus

function with ε(h) → 0 as h → 0. Let F hM denote a monotone scheme and F h

A an accurate scheme.

The filtered scheme is defined as

F h[u] = F hM [u] + ε(h)S

(

F hA[u] − F h

M [u]

ε(h)

)

.

We start by showing that the filtered schemes are nearly monotone.

Proposition 3.4.8. A filtered scheme is nearly monotone.

Proof. We can write the filtered scheme as

F h[u] = F hM [u] + F h

N [u]

where

F hN [u] = ε(h)S

(

F hA[u] − F h

M [u]

ε(h)

)

.

Since the filter function S is bounded, F hN is a perturbation and therefore F h is a nearly

monotone scheme.

Stability of the filtered schemes follows from the stability of the underlying monotone

scheme, the continuity of the filter function and Proposition 3.4.5. Consistency is a con-

sequence of the nearly monotonicity and the consistency of the underlying monotone

scheme. The convergence of the filtered schemes then follows from Theorem 3.4.3.

Remark 3.12. It is important to notice that there are no requirements on the accurate scheme:

it can simply be constructed with standard higher order finite differences, or it can be

designed to take advantage of known properties of the solutions to the equation under

consideration.

Finally, we explain how filtered schemes can achieve higher accuracy: the parameter

ε(h) must be chosen carefully. Heuristically, it should be large enough to allow the accurate


scheme to be active where the solution is smooth, and small enough to force the monotone

scheme to be active when the solution is singular (in order to guarantee convergence).

Proposition 3.4.9. Suppose that the formal discretization errors of the schemes F hM , F h

A are

O(hβM ) and O(hβA), respectively. Choose α such that βA > βM > α > 0. Then if φ smooth and

ε(h) = O(hα), then F h[φ] = F hA[φ] for sufficiently small h.

Proof. If φ is smooth, then

F hA[φ] − F h

M [φ]

ε(h)=

O(hβA) + O(hβM )

O(hα)= O(hβM −α) ≤ O(1)

for sufficiently small h. Hence,

F h[φ] = F hM [φ] + ε(h)S

(

F hA[φ] − F h

M [φ]

ε(h)

)

= F hM [φ] + F h

A[φ] − F hM [φ] = F h

A[φ]

since S is equal to the identity in a neighbourhood of the origin by definition of filter

function.

In summary, filtered schemes combine a stable, monotone, consistent scheme with

an accurate (but possibly unstable) scheme. The accurate scheme is not required to be

stable on its own. However, independently of the choice made, the combination of the two

schemes is both provably convergent, and (potentially) higher order accurate.

CHAPTER 4

FILTERED SCHEMES FOR HAMILTON-JACOBI EQUATIONS

4.1 INTR ODUC TI ON

In this chapter, we build filtered schemes for first order Hamilton-Jacobi (HJ) partial

differential equations, with special focus on the Eikonal equation. These form a simple and

general class of finite difference schemes that combine a stable, monotone scheme with

an accurate (but possibly unstable) scheme. By construction they are nearly monotone

(see Definition 3.4.2) and thus provably convergent to the unique viscosity solution of the

underlying equation (see Theorem 3.4.3). Moreover, they have the potential to achieve

higher accuracy (when compared to monotone schemes) by making a careful choice of the

filtering mechanism (see Proposition 3.4.9).

We take a particular interest on the eikonal equation

|∇u(x)| = f(x), for x outside Γ,

u(x) = g(x), for x on Γ.(4.1)

where f > 0 and Γ is here a closed, bounded set. We consider as well HJ equations of the

form

H(x, ∇u) = f(x), x ∈ Ω,

u(x) = g(x), x ∈ Γ,(4.2)

where ∇u is the gradient of the function u, Ω is an open set, Γ is the boundary of Ω and

the Hamiltonian H is a nonlinear Lipschitz continuous function. We always refer to the

eikonal equation specifically, even though it is in fact an HJ equation (take H(p) = |p|).When we refer to HJ equations we always have more general equations in mind.

In general, solutions are not smooth (or even differentiable) and so we consider viscosity

solutions (see chapter 2). The viscosity solutions can be piecewise smooth with a singularity

in the gradient. It therefore makes sense to design high order schemes that provide higher

order accuracy (at least) away from these singularities.

4.1.1 Contribution of this work

We build filtered schemes for first order HJ equations. As discussed in section 3.4, filtered

schemes allow for a wide choice of accurate schemes, which are not required to be stable

39

40 4.2. DISCRETIZATION AND SOLVERS

on their own. Here we exploit this and choose accurate schemes that are designed to take

advantage of known properties of the solutions of HJ equations. The main contribution

is in the choice of the filter function together with a judicious choice of the accurate

scheme. We show that using one-sided higher order finite differences for the accurate

scheme, combined with an upwind monotone scheme results in a very simple, explicit and

accurate scheme for the eikonal equation. We treat as well the general case of first order HJ

equations. In summary, the schemes we introduce have the following properties.

1. They are simple and easy to implement on Cartesian grids. For example, for the

eikonal equation the filtered scheme using the centered difference scheme, is conver-

gent and examples show that second order accuracy is obtained, which results in the

simplest second order accurate finite difference scheme.

2. Higher order explicit schemes are obtained using higher order upwind interpolation.

These higher order schemes can be solved using fast sweeping.

3. Other choices of accurate schemes can be used instead: we implement ENO schemes

for comparison. Any choice of discretization (e.g. the popular discontinuous Galerkin

method) can be used, provided a monotone scheme can also be constructed in the

same setting.

4. For the eikonal equation in one dimension, higher order convergence for the numeri-

cal solution is proved, even for non-smooth solutions.

5. For HJ equations (in general), higher order convergence is obtained locally, in regions

where the solution is smooth.

4.2 D IS C RETIZATION AND S OLVER S

In this section we will discuss the discretization of the monotone and filtered schemes

for both HJ and eikonal equations for different choices of the accurate schemes (centered,

upwind and ENO). We do this both in one and two dimensions. We should point out

that all discretizations for HJ equations can be applied to the eikonal equation, although

we choose to present and use specific discretizations for the eikonal equation given its

importance in the literature.

We consider only the case of regular Cartesian grids since the discretization is simpler

and the idea is clear. It is certainly possible to build filtered schemes using higher order

methods on triangulated grids for example.

Chapter 4. Filtered Schemes for Hamilton-Jacobi equations 41

4.2.1 Monotone schemes

For the eikonal equation, in the one-dimensional case, the monotone scheme is given by

∣

∣

∣uhx

∣

∣

∣

M= max

−u(x + h) − u(x)

h,u(x) − u(x − h)

h, 0

. (4.3)

Since we are working on a Cartesian grid, extending it to the two dimensional case simply

requires the use of the standard Euclidean 2-norm function N : R2 → R given by

N(x, y) =√

x2 + y2. (4.4)

We then define∣

∣

∣∇uh∣

∣

∣

M= N

(

∣

∣

∣uhx

∣

∣

∣

M,∣

∣

∣uhy

∣

∣

∣

M)

(4.5)

which is monotone as desired (see Examples 3.4 and 3.5).

There are several monotone numerical Hamiltonians we could use to discretize HJ equa-

tions. Here we choose to use the Lax-Friedrichs numerical Hamiltonian [KOQ04], because

it has a simple form and it can be used for both convex and nonconvex Hamiltonians:

HhLF [u](x) = Hh

LF (x, p+, p−) = H

(

x,p+ + p−

2

)

− σxp+ − p−

2(4.6)

where σx is the artificial viscosity satisfying σx = max∣

∣

∣

∂H∂p

∣

∣

∣, p = ux and p± are the corre-

sponding forward and backward differences approximations of ux.

The scheme easily generalizes into higher dimensions: in the two-dimensional case we

have

HhLF [u](x, y) = Hh

LF (x, y, p+, p−, q+, q−)

= H

(

x, y,p+ + p−

2,q+ + q−

2

)

− σxp+ − p−

2− σy

q+ − q−

2

(4.7)

where σy = max∣

∣

∣

∂H∂q

∣

∣

∣, q = uy and q± are the corresponding forward and backward differ-

ences approximations of uy.

4.2.2 Accurate schemes

We know that the filtered scheme will converge independently of the choice of the accurate

scheme. Its purpose is to provide additional accuracy in the regions where the solution is

smooth and where the accurate scheme is active. Thus the resulting accuracy of the solution

comes from a judicious choice of the accurate scheme. In addition to the accuracy, the


choice of accurate scheme determines the type of solver we can use (iterative or sweeping),

based on whether an explicit solution formula is available (see subsection 4.2.5).

We first consider the one-dimensional case and then show how, as in the previous

section, the schemes can be generalized for the two-dimensional case.

Centered Schemes: The second order accurate centered scheme are obtained by simply

replacing ux by its second order centered approximation:

∣

∣

∣uhx

∣

∣

∣

C,2=

|u(x + h) − u(x − h)|2h

,

HhC,2[u](x) = H

(

x,u(x + h) − u(x − h)

2h

)

.

Upwind Schemes: The upwind schemes proposed here were first thought for the

eikonal equation, although they can be generalized to HJ equations. In the eikonal equation

case, they are designed to choose the finite difference stencil in terms of the direction of

the characteristics of the solution. This means using the left (right) biased stencil if the

characteristics are being propagated from the left (right). The higher order upwind schemes

generalize the monotone scheme above. They are defined as follows.

Set P ±,n[u] to be the interpolating polynomial of degree n of u at the nodes xj = x ± jh

for j = 0, 1, . . . , n. (The sign in the superscript indicates interpolation to the left or to

the right.) These interpolating polynomials are standard and given in several convenient

explicit forms (see [Ise09]). We give a specific example below. We then set

∣

∣

∣uhx

∣

∣

∣

U,n= max

− d

dxP +,n[u](x),

d

dxP −,n[u](x), 0

,

HhU,n[u](x) = Hh

LF

(

x,d

dxP +,n[u](x),

d

dxP −,n[u](x)

)

.

ENO Schemes: High order essentially non-oscillatory (ENO) are another option for

the accurate discretization. (A refinement of ENO is WENO [JP00], which we choose not

to implement, since the main idea is clear from the ENO examples.) The idea underlying

the ENO schemes is to do a standard interpolation using an adaptive stencil, i.e., the

stencil used depends on the function being interpolated. Starting with two nodes, the ENO

interpolation of order n selects the remaining n − 1 interpolation nodes by successively

adding nodes to the stencil with the smallest Newton divided difference. This way, the rth

node is chosen by comparing two approximations of the derivative of order r + 1, with r

taking successively the values 1, . . . , n − 1.

Let En,± 12 [u] denote the ENO interpolation as explained above, and as defined in [OS91].


Then we define the nth-order accurate ENO scheme to be

∣

∣

∣uhx

∣

∣

∣

E,n= max

− d

dxEn, 1

2 [u](x),d

dxEn,− 1

2 [u](x), 0

,

HhE,n[u](x) = Hh

LF

(

x,d

dxEn, 1

2 [u](x),d

dxEn,− 1

2 [u](x)

)

.

Two dimensional schemes. In the case of the eikonal equation we use (4.4) as we did

in subsection 4.2.1. The second order centered scheme becomes

∣

∣

∣∇uh∣

∣

∣

C,2= N

(

∣

∣

∣uhx

∣

∣

∣

C,2,∣

∣

∣uhy

∣

∣

∣

C,2)

, (4.8)

the upwind schemes become

∣

∣

∣∇uh∣

∣

∣

U,n= N

(

∣

∣

∣uhx

∣

∣

∣

U,n,∣

∣

∣uhy

∣

∣

∣

U,n)

, (4.9)

and, finally, the ENO schemes are defined as

∣

∣

∣∇uh∣

∣

∣

E,n= N

(

∣

∣

∣uhx

∣

∣

∣

E,n,∣

∣

∣uhy

∣

∣

∣

E,n)

. (4.10)

The upwind schemes defined here for the eikonal equation recover the 2nd and 3rd order

upwind schemes from ([Set99a], [Cho01] and [ABM+11]). These schemes have been solved

in the literature using Fast Marching algorithms.

As for HJ equations, the extension to two dimensions follows from using the two-

dimensional expression of HhLF as we did with the monotone scheme.

4.2.3 Boundary conditions

In this section we discuss the treatment of boundary conditions for the filtered scheme.

First we discuss the one dimensional case. Note that we solved the internal problem

and so the Dirichlet data is prescribed on the boundary of the computational domain. For

the monotone difference method this leads to a standard application of Dirichlet boundary

conditions.

For higher order accurate methods, the situation is similar to the case of multistep

methods for ordinary differential equations: more information is needed to achieve the

higher accuracy. This information can take the form of additional function values at adja-

cent grid points, or higher derivative information [HNW93]. For practical considerations,

in order to test the accuracy of the solution without introducing errors from the boundary,

we extend the Dirichlet data to more grid points. More precisely, we set the exact solution

(in fact, an nth order approximation of the exact solution is enough) at the n grid points


adjacent to the boundary when using the nth order upwind and ENO filtered schemes.

If the additional information is not available we may lose the higher accuracy. Using

just the first order accurate monotone scheme reduces the order of the global accuracy.

Similarly, using only the available one sided higher order approximations may decrease

the accuracy since the available direction is not the one we are interested in: as we will see

below in the proof of Theorem 4.2.4 in subsection 4.2.7 for the eikonal equation case, we

want to interpolate towards the boundary and not away from it.

In the two-dimensional case, for the eikonal equation, we solved the external problem

and so the boundary of the computational domain did not include the Dirichlet boundary.

This poses an additional difficulty since the schemes need to be carefully defined near the

boundary of the computational domain to prevent computational errors that propagate into

the computational domain. Here we dealt with this issue as is usually done for monotone

schemes: we consider only the one sided differences available. Since the characteristics

go inward, the lack of external information is not a problem. For the (internal) Dirichlet

boundary, we proceed in the same way we did in the one-dimensional case: we set the

exact solution at as many adjacent grid points of the boundary as needed depending on

the order of accuracy of the scheme used.

For general HJ equations, the computational boundary can cause problems, depending

on the discretization used. For the Godunov scheme, which reduces to (4.5) in the case

of the eikonal equation, there are no problems, so this is what we used for the eikonal

equation. However, for general HJ equations in two dimensions using the Lax-Friedrichs

schemes (4.7) with high order interpolation is more complicated [ZZQ06], and can lead to

errors at the computational boundary.

4.2.4 Filtered schemes

We can now define the filtered schemes, which were discussed in detail in section 3.4. Let

F hM denote the monotone discretization of the operator on the grid with spacing h, given

above in subsection 4.2.1. Let F hA denote an accurate discretization of the same operator,

with several possible choices being given above in subsection 4.2.2.

Here we define the filtered scheme using the discontinuous filter function of Exam-

ple 3.7 and taking ε(h) =√

h. This leads to the filtered scheme, F h, given by the following

simple formula:

F h[u] =

F hA[u], if

∣

∣

∣F hA[u] − F h

M [u]∣

∣

∣ ≤√

h,

F hM [u], otherwise.

(4.11)


Remark 4.1. The choice of the factor√

h in (4.11) is designed to fit between two rates: large

enough to permit the accurate scheme to be active where the solution is smooth (see

Proposition 3.4.9) , and small enough to force the monotone scheme to be active when the

solution is singular. In the case of the eikonal equation, the monotone scheme is accurate

to O(h) and the accurate scheme is O(h2) or better, thus our choice of the factor√

h.

Below, at the end of subsection subsection 4.3.1, we consider an example where the

Hamiltonian is non-convex, and the observed convergence rate is O(√

h) for the monotone

scheme, and so we take the factor to be smaller than√

h.

Theorem 4.2.1 (Convergence of the filtered schemes for HJ equations). Consider the Dirichlet

problem (4.2) on a bounded domain Ω and assume that it has a strong comparison principle (see

Definition 2.3.7). Let F h be the filtered scheme (4.11) and Uh ∈ C(Gh) be any of its solutions. If

F h is stable then

limh→0

uh(x) = u(x), for all x ∈ Ω


Proof. The filtered scheme F h is consistent since both underlying schemes, F hM and F h

A, are

consistent and is nearly monotone as a consequence of Proposition 3.4.8. Thus, since F h is

stable by assumption, the convergence follows from Theorem 3.4.3.

To apply the theorem, we need to show that the filtered scheme is stable. Stability will

follow from the well-posedness of the monotone schemes and the continuity of the filter

function (see Proposition 3.4.5 and Remark 3.11).

In our setting, although discontinuous, (4.11) has a simple form which allows for

explicit solution formulas as we will see in subsection 4.2.5. These explicit solution

formulas allow us to build fast sweeping solvers, which are appropriate for HJ equations.

In practice the computational results are as good as could be expected. For the purpose of

the proof, a continuous filter is needed but the practical advantages of the discontinuous

one outweigh the lack of rigor.

Remark 4.2. In [FO13], a continuous interpolation between the monotone and accurate

scheme was used. There, this was required to show the stability of the filtered schemes,

but it was also of practical use for a Newton solver.

Theorem 4.2.1 does not provide any information regarding the convergence rate. Prov-

ing higher order convergence requires additional efforts and is possible in specific settings.

For the one-dimensional eikonal equation, we prove higher order convergence in subsec-

tion 4.2.7. For the two-dimensional eikonal equation, second and third order convergence

is proven for smooth solutions in [ABM+11]. We are more interested in demonstrating


the higher order convergence in practice, which is done using numerical simulations. In

particular, in the case of piecewise smooth solutions in two dimensions, we achieve second

order convergence rates in the smooth region, and first order convergence overall in the l∞

norm.

Remark 4.3. In addition to stationary equations, we can build filtered schemes for time

dependent equations. This can be accomplished by using the filtered scheme on the spatial

part of the operator, and a standard time discretization (forward Euler or strong stability

preserving time discretizations [GKS11]) for the time derivative. As needed, the filter could

also be applied to the time derivative term as well. In this case, with minor modifications,

the proof of convergence for the filtered scheme goes through, since, as it is standard for

viscosity solution, the time derivative can be considered as an additional spatial variable.

Following ideas similar to these, filtered schemes for time dependent equations were built

for first order HJ equations [BFS16], second order Hamilton-Jacobi-Bellman equations

[BPR16] and front propagation [Sah16].

4.2.5 Explicit methods

For upwind schemes, the interpolation is fixed, so we can solve for the reference variable

and build explicit schemes. In contrast, it is difficult to directly build explicit methods for

many of the other schemes. Rather than present the general method for solving for the

reference variable and in order to be concrete (and save space), we give a specific example

below. The general method should then be clear.

Eikonal equations

Example 4.1 (one-dimensional case). Consider first the monotone scheme in the one-

dimensional case (4.3). Solving the equation∣

∣

∣uhx

∣

∣

∣

M= f for the reference variable, u(x),

leads to

u(x) = minu(x + h), u(x − h) + hf(x) (4.12)

since f > 0. Consider now the second order upwind scheme, again in one dimension. The

upwind scheme takes the form

∣

∣

∣uhx

∣

∣

∣

U,2 ≡ 1

2hmax 3u(x) − 4u(x ± h) + u(x ± 2h), 0 = f.

Solving the preceding equation for the reference variable, u(x), leads to

u(x) =1

3min4u(x + h) − u(x + 2h), 4u(x − h) − u(x − 2h) +

2

3hf(x). (4.13)


Finally, consider the correspondent filtered scheme. Combining (4.12) and (4.13) and

using the definition of the filtered scheme (4.11) we obtain the following explicit represen-

tation of the solution of the filtered scheme at a reference point in terms of the neighboring

values

u(x) =

1

3min4u(x ± h) − u(x ± 2h) +

2

3hf(x) if

∣

∣

∣

∣

∣

∣

∣uhx

∣

∣

∣

A −∣

∣

∣uhx

∣

∣

∣

M∣

∣

∣

∣

≤√

h,

minu(x + h), u(x − h) + hf(x) otherwise.

Example 4.2 (two-dimensional case). We can also obtain an explicit solution for the filtered

schemes using the upwind scheme in the two-dimensional case as above. In this case

solving for the reference variable u(x, y) requires solving a nonlinear equation of the form

[

(z − a)+]2

+[

(z − b)+]2

= c2

for the unknown z where a, b and c > 0 are constants and (z)+ := maxz, 0. This equation

combines piecewise linear functions with a quadratic function. The unique solution of the

equation is given by

z =

mina, b + c |a − b| ≥ c,

a + b +√

2c2 − (a − b)2

2|a − b| < c,

(4.14)

(see e.g. [Zha05] for a derivation).

In the case of the monotone scheme we get

a = minu(x + h, y), u(x − h, y),

b = minu(x, y + h), u(x, y − h),

c = hf(x).

As for the second order upwind scheme we have

a =1

3min4u(x ± h, y) − u(x ± 2h, y),

b =1

3min4u(x, y ± h) − u(x, y ± 2h),

c =2

3hf(x).

The explicit formula of the filtered scheme can then be obtained as in the one-dimensional

case using the definition of filtered scheme (4.11) and (4.14).


Hamilton-Jacobi equations

Example 4.3 (one-dimensional case). Consider first the monotone scheme (4.6). We know

that

p+ =u(x + h) − u(x)

h, p− =

u(x) − u(x − h)

h.

Thus, solving HhLF [u] = f for the reference variable, u(x), leads to

u(x) =h

σx

[

f(x) − H

(

x,u(x + h) − u(x − h)

2h

)

+ σxu(x + h) + u(x − h)

2h

]

.

Consider now the second order upwind scheme. We have

d

dxP +,2[u](x) =

−3u(x) + 4u(x + h) − u(x + 2h)

2h,

d

dxP −,2[u](x) =

3u(x) − 4u(x − h) + u(x − 2h)

2h.

Thus, solving HhU,2[u] = f for the reference variable, u(x), leads to

u(x) =2h

3σx

[

f(x) − H

(

x,−u(x + 2h) + 4u(x + h) − 4u(x − h) + u(x − 2h)

4h

)

+σx−u(x + 2h) + 4u(x + h) + 4u(x − h) − u(x − 2h)

4h

]

.

The explicit formula of the filtered scheme can then be obtained as in the eikonal

equation case using the definition of filtered scheme (4.11).

For the ENO schemes, we can’t get an explicit formula. However, it is possible to get a

fixed point iteration which was used successfully with a fast sweeping solver in [ZZQ06].

4.2.6 Solution methods

The simplest solver is to use the fixed point iteration

un+1 = un − dt(F h[u] − f) (4.15)

which corresponds to the discrete version of the parabolic equation ut + (F [u] − f) = 0

using a forward Euler step, where F [u] = |∇u| or F [u](x) = H(x, ∇u). The fixed point

iteration will be a contraction in the l∞ norm provided that we choose dt small enough

as dictated by the nonlinear CFL condition [Obe06b], which in the eikonal equation case

means dt = O(h) (see section 3.2 for more details). This will however make the solver

relatively slow.


As seen in the previous section, we have explicit formulas for the upwind filtered

schemes. This allows us to use the fast sweeping method [TCOZ03, Zha05], which is a

fast iterative solution method. Each node is updated using Gauss-Seidel iterations with

alternating sweeping ordering of the domain. This allows information to propagate from Γ

along characteristics to the rest of the computational domain. In the case of the eikonal

equations, an alternative would be the Fast Marching Method [Set99a, Tsi95]: the solution

is constructed by using characteristic information to select the next node where the solution

can be obtained. However this requires a complicated data structure which makes it more

difficult to implement. In one dimension, the whole domain is swept with two alternating

ordering of the nodes

• (i = 1, . . . , N) and (i = N, . . . , 1)

which correspond to the two possible directions for the propagation of the characteristics.

In two dimensions we sweep the whole domain with eight alternating ordering of the

nodes

• (i = 1, . . . , N, j = 1, . . . , N),

• (i = 1, . . . , N, j = N, . . . , 1),

• . . .

• (j = N, . . . , 1, i = N, . . . , 1).

corresponding respectively to up-right, up-left, down-left, down-right, right-up, left-up,

left-down and right-down. Here, the first (last) four orderings speed up the convergence

when the characteristics are aligned with the x-axis (y-axis).

For the filtered centered and ENO schemes, we implemented the fixed point solver

(4.15). For the upwind filtered schemes we implemented the fast sweeping solver described

above.

4.2.7 Error estimates in one dimension

In this section, we focus on the eikonal equation in one dimension, with Dirichlet boundary

conditions on the endpoints of an interval. Despite the fact that the solution is Lipchitz

continuous, we are able to prove, when the data f is smooth enough, that the upwind

schemes converge to higher order. This is a consequence of the fact that (i) the solution

is piecewise smooth, and we can express it as a minimum of the two ODE solutions (ii)


the numerical solution is also expressed as the minimum of the left and right branches. A

similar idea was used to obtain higher accuracy for conservation laws in [EFT13].

Here we prove the higher order convergence of a particular scheme: the (unfiltered)

high order upwind schemes. In this case we do not prove convergence of the filtered

scheme which combined the high order upwind scheme with the monotone upwind

scheme. However, we implement the filtered scheme, and we found, in practice, for the

computed solution, the higher order scheme is always active.

Remark 4.4. The reason for using the filtered scheme is that it provides global stability:

intermediate numerical solutions are stable, even though in the final computed solution

the accurate scheme is always active. To use a simile, the filtered scheme acts like training

wheels on a bicycle, maintaining stability even though, ultimately the training wheels do

not touch the ground.

We consider u to be the viscosity solution of the one-dimensional eikonal equation

|u′(x)| = f(x), x ∈ (a, b),

u(x) = g(x), x ∈ Γ = a, b.(4.16)

To start we need first to recall the known Dynamic Programming Principle (DPP).

Proposition 4.2.2. Consider the dynamics

y(t) = α(t) t ∈ (0, +∞),

y(0) = x,

and cost functional

Jx(α(·)) =∫ tx(α)

0f(yx(s; α)ds + g(yx(tx(α), α)),

where A = α(·) : [0, +∞) → −1, 1 ⊂ R, measurable and tx denotes the entry time in Γ.

Hence u is the value function of a minimum cost problem, being given by

u(x) = infα∈A

Jx(α(·)). (4.17)

Proof. See [BCD97, Chapter IV].

We are now able to express u as the minimum of two ODE solutions.

Proposition 4.2.3. The viscosity solution u of (4.16) is given by

u(x) := minua(x), ub(x), (4.18)


where ua and ub are respectively the solution of

u′(x) = f(x),

u(a) = g(a),and

−u′(x) = f(x),

u(b) = g(b).(4.19)

Proof. Since f > 0, the only trajectories to be considered in the minimum of (4.17) are

the ones that travel straight to the endpoints a and b. These trajectories are given by the

controls α1 ≡ −1 and α2 ≡ 1, respectively. Hence

u(x) = min Jx(α1(·)), Jx(α2(·)) .

It is now easy to see that ua(x) = Jx(α1(·)) and ub(x) = Jx(α2(·)) and so we are done.

We can now prove our result.

Theorem 4.2.4. For n ≤ 6 and if f ∈ C(n+1)[a, b] the upwind schemes are convergent. Moreover,

if the solution is denoted by uh,n, we have the following error estimate

∣

∣

∣uh,n(a + jh) − u(a + jh)∣

∣

∣ ≤ ChnMn+1 (4.20)

for j = 0, . . . , b−ah

, where C is a constant depending on n, the Lipschitz constant of f , a and b and

Mn = maxx∈[a,b]

∣

∣

∣f (n)(x)∣

∣

∣.

Proof. The idea of the proof consists in solving (4.19) with backward difference schemes

and realize using (4.18) that we recover uh,n, more precisely, the explicit formulas for

upwind schemes discussed in subsection 4.2.5. The assumption n ≤ 6 is needed as the

backward schemes are only stable (and therefore convergent) when n ≤ 6.

Let uh,na and uh,n

b denote respectively the solutions obtained using backward schemes to

solve (4.19). Hence they are the solution of

U−,n[u](x) = f(x)

u(a + jh) given for j = 0, . . . , n − 1,

−U+,n[u](x) = f(x)

u(b − jh) given for j = 0, . . . , n − 1.

Set uh,n(x) := minuh,na , uh,n

b . Under our assumptions we know that uh,na and uh,n

b converge

respectively to ua and ub (see [QSS07] on multistep methods). Therefore the proof is done

if we show that uh,n(x) = uh,n(x).

Rather than prove this for all n, we give a particular example (n = 2) and the general

case should then follow easily. We will use the second order backward differentiation

schemes and will therefore recover the second order upwind schemes. We have that uh,2a is

52 4.3. COMPUTATIONAL RESULTS

the solution of3u(x) − 4u(x − h) + u(x − 2h)

2h= f(x)

and can therefore be written as

uh,2a (x) =

1

3(4u(x − h) − u(x − 2h)) +

2h

3f(x).

Likewise, uh,2b is the solution of

−−3u(x) + 4u(x + h) − u(x + 2h)

2h= f(x)

and so

uh,2b (x) =

1

3(4u(x + h) − u(x + 2h)) +

2h

3f(x)

Using now (4.18), we recover (4.13) as desired.

Thus the accuracy of the numerical solution of (4.16) is determined by the accuracy of

the numerical solution of each of the two linear odes (4.19).

The error estimates result naturally from the error estimates for backward difference

schemes for ODEs which can be found in [QSS07].

Remark 4.5. The requirement f ∈ C(n+1)[a, b] is needed to obtain the order of convergence.

This requirement can be relaxed to f being piecewise C(n+1) in the same regions as the

solution u. The idea is that we only need uh,na and uh,n

b to be high order convergent when

they are active in the minimum of (4.18).

Remark 4.6. Here we assume the exact solution is known near the boundary, but this

assumption can be relaxed. The same order of accuracy can be obtained provided the

boundary conditions are known to sufficient precision near the boundary, i.e., with the

same of order of accuracy. Furthermore, these can be computed from the boundary data

using standard methods [QSS07].

4.3 COMP UTATIONAL RES ULTS

4.3.1 Example solutions in one dimension

In this subsection we discuss the examples considered in one dimension. In all of them

the solution is piecewise smooth with a single singularity. Their purpose is confirm the

improved accuracy of the filtered schemes, as well as the high order convergence of the

upwind schemes for the eikonal equation. All examples are displayed in Figure 4.1.

The first example is the eikonal equation with f(x) = 1 + cos(x) with the Dirichlet

boundary conditions being prescribed at x = ±2. The computational domain is [−2, 2].


−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

1

2

3

4

5

6

7

8

9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

−2 −1.5 −1 −0.5 0 0.5 1 1.5 214.5

15

15.5

16

16.5

17

17.5

18

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 4.1: Profile of the solutions of the five examples considered in one dimension (at the top, eikonalequation examples, at the bottom, HJ equations examples).

The exact solution is given by u(x) = 3 − |x + sin(x)| and it is therefore piecewise smooth

with a singularity at x = 0 (see Figure 4.1). We represent the solution obtained with the

monotone scheme and the 2nd order upwind filtered scheme for 50 mesh points near the

singularity in [−0.4, 0.4] on Figure 4.2.

The second example is again the eikonal equation with f(x) = 1 + e|x|, where the

Dirichlet boundary conditions are once again prescribed at x = ±2 and the computational

domain is [−2, 2]. The exact solution is given by u(x) = 10−|x|−e|x| and as in the previous

example, it is piecewise smooth with a singularity at x = 0 (see Figure 4.1).

The third example, also a solution of the eikonal equation, is given by

u(x) =

x3 + ax x ∈ [0, x0],

1 + a − ax − x3 x ∈ [x0, 1],

with a =1−2x3

0

2x0−1, x0 =

3√2+2

4 3√2and therefore f(x) = 3x2 + a. This example was chosen for two

main reasons: there is no symmetry in the relationship between the singularity and the

grid points, as opposed to the two previous examples where the singularity was always

a midpoint of two consecutive grid points; this is one the examples in [Abg09] that the

author uses to check the rate of convergence of the proposed method. The difference is that

in [Abg09] the error in the l∞ norm is computed at the grid points in the interval[

13√2

, 12

]

,


−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.42.5

2.55

2.6

2.65

2.7

2.75

2.8

2.85

2.9

2.95

3

Monotone

2ndupwind

Exact

Figure 4.2: Exact solution and solutions obtained with the monotone scheme and the 2nd order upwindfiltered scheme with 50 grid points for the first example of the eikonal equation.

instead of all the grid points as we do here. The author chooses that interval since it is an

interval where the solution is smooth but as we explained above we can look at the error

on all grid points and still obtain the high order convergence.

We consider as well two HJ equations. The first one given by H(p) = p2, a convex

Hamiltonian, with f(x) = ex and

u(x) =

−2ex2 + 20 x ∈ [−2, 0],

2ex2 + 16 x ∈ [0, 2].

The second one given by H(p) = cos(p)2 + |p|, a nonconvex Hamiltonian considered in

[Abg09], with u(x) = e−|x| and f(x) = cos(e−|x|)2 + e−|x|. The profile of both solutions is

depicted in Figure 4.1 and the Dirichlet boundary conditions are prescribed at x = ±2,

with the computational domain being [−2, 2]. For the nonconvex example, the factor√

h

in the filtered scheme (4.11) was replaced by h1

10 (see Remark 4.1).

The computational domain is discretized on a grid with N points and the singularity is

never a grid point.


4.3.2 Computational results in one dimension

In this subsection we discuss the computational results obtained in one dimension. The

main purpose is to demonstrate that the filtered scheme achieves the higher order accuracy

and that, in particular for the eikonal equation, the upwind filtered schemes achieve higher

order convergence rate as proved in subsection 4.2.7 for the (unfiltered) upwind schemes.

We organize the discussion in three parts: accuracy and behaviour, order of convergence

and upwind vs ENO. For the eikonal equation, we obtained results with the monotone

scheme (4.3) and the respective filtered schemes using as the accurate scheme the second

centered scheme and the second, third and forth order upwind and ENO schemes. For HJ

equations, we obtain results using the monotone scheme (4.6) and the respective filtered

schemes using as the accurate scheme the second order centered, upwind and ENO

schemes. Third order upwind and ENO filtered schemes were also used, but they did not

show any advantage over the second order schemes.

Errors and order, 1st ExampleN Monotone 2nd Upwind 3rd Upwind 4th Upwind64 4.465 × 10−2 - 1.141 × 10−3 - 8.532 × 10−5 - 2.646 × 10−6 -

128 2.223 × 10−2 0.99 2.908 × 10−4 1.95 1.076 × 10−5 2.95 1.700 × 10−7 3.92256 1.109 × 10−2 1.00 7.337 × 10−5 1.98 1.348 × 10−6 2.98 1.074 × 10−8 3.96512 5.538 × 10−3 1.00 1.842 × 10−5 1.99 1.687 × 10−7 2.99 6.745 × 10−10 3.981024 2.767 × 10−3 1.00 4.615 × 10−6 1.99 2.109 × 10−8 3.00 4.224 × 10−11 3.99

N 2nd centered 2nd ENO 3rd ENO 4th ENO64 6.553 × 10−4 - 7.660 × 10−4 - 2.780 × 10−5 - 5.561 × 10−7 -

128 1.559 × 10−4 2.05 1.918 × 10−4 1.97 3.546 × 10−6 2.94 3.544 × 10−8 3.93256 3.789 × 10−5 2.03 4.803 × 10−5 1.99 4.470 × 10−7 2.97 2.236 × 10−9 3.96512 9.451 × 10−6 2.00 1.201 × 10−5 1.99 5.608 × 10−8 2.99 1.404 × 10−10 3.981024 2.317 × 10−6 2.03 3.004 × 10−6 2.00 7.022 × 10−9 2.99 8.776 × 10−12 3.99

Table 4.1: Accuracy in the l∞ norm and order of convergence of the schemes for the first example of theeikonal equation.

Accuracy and behaviour of the filtered schemes.

We begin by comparing the accuracy of the monotone scheme with the filtered schemes

by looking at the error in the l∞ norm in Figure 4.5 and Tables 5.3, 5.4, 5.5, 5.6, 5.7. As

expected the filtered schemes have improved accuracy.

Once close to the solution, the filtered schemes behave as designed choosing to use

the accurate scheme whenever possible, i.e., whenever they interpolate the solution in a


Errors and order, 2nd ExampleN Monotone 2nd Upwind 3rd Upwind 4th Upwind64 1.997 × 10−1 - 8.011 × 10−3 - 3.642 × 10−4 - 1.766 × 10−5 -

128 9.984 × 10−2 0.99 2.042 × 10−3 1.95 4.716 × 10−5 2.92 1.162 × 10−6 3.88256 4.992 × 10−2 0.99 5.153 × 10−4 1.98 5.995 × 10−6 2.96 7.441 × 10−8 3.94512 2.496 × 10−2 1.00 1.294 × 10−4 1.99 7.555 × 10−7 2.98 4.706 × 10−9 3.971024 1.248 × 10−2 1.00 3.242 × 10−5 1.99 9.482 × 10−8 2.99 2.959 × 10−10 3.99


128 1.570 × 10−3 2.00 1.018 × 10−3 1.95 2.764 × 10−4 2.60 1.823 × 10−3 -0.29256 3.859 × 10−4 2.01 2.573 × 10−4 1.97 4.899 × 10−5 2.48 2.499 × 10−4 2.85512 9.700 × 10−5 1.99 6.466 × 10−5 1.99 8.981 × 10−6 2.44 5.037 × 10−5 2.301024 2.410 × 10−5 2.01 1.621 × 10−5 1.99 1.422 × 10−6 2.66 4.332 × 10−5 0.22

Table 4.2: Accuracy in the l∞ norm and order of convergence of the schemes for the second example of theeikonal equation.

Errors and order, 3rd ExampleN Monotone 2nd Upwind 3rd Upwind 4th Upwind64 1.368 × 10−2 - 3.079 × 10−4 - 1.332 × 10−15 - 2.887 × 10−15 -

128 6.756 × 10−3 1.01 7.860 × 10−5 1.95 1.110 × 10−15 0.26 3.109 × 10−15 -0.11256 3.417 × 10−3 0.98 1.960 × 10−5 1.99 2.220 × 10−15 -0.99 4.219 × 10−15 -0.44512 1.703 × 10−3 1.00 4.924 × 10−6 1.99 2.887 × 10−15 -0.38 2.665 × 10−15 0.661024 8.521 × 10−4 1.00 1.232 × 10−6 2.00 5.107 × 10−15 -0.82 4.663 × 10−15 -0.81


128 1.596 × 10−4 1.97 7.860 × 10−5 1.95 7.550 × 10−15 -1.61 4.441 × 10−15 -1.72256 3.950 × 10−5 2.00 1.960 × 10−5 1.99 2.109 × 10−14 -1.47 3.819 × 10−14 -3.09512 9.886 × 10−6 1.99 4.924 × 10−6 1.99 3.220 × 10−14 -0.61 1.134 × 10−13 -1.57

1024 2.192 × 10−6 2.17 1.232 × 10−6 2.00 5.818 × 10−14 -0.85 8.527 × 10−14 0.41

Table 4.3: Accuracy in the l∞ norm and order of convergence of the schemes for the third example of theeikonal equation.

Errors and order, 4th ExampleN Monotone 2nd centered 2nd Upwind 2nd ENO64 1.234 × 10−1 - 8.532 × 10−2 - 9.307 × 10−2 - 8.308 × 10−2 -128 6.106 × 10−2 1.00 4.226 × 10−2 1.00 4.179 × 10−2 1.14 4.132 × 10−2 1.00256 3.037 × 10−2 1.00 2.108 × 10−2 1.00 2.095 × 10−2 0.99 2.067 × 10−2 0.99512 1.515 × 10−2 1.00 1.054 × 10−2 1.00 1.044 × 10−2 1.00 1.057 × 10−2 0.97

1024 7.563 × 10−3 1.00 5.310 × 10−3 0.99 5.304 × 10−3 0.98 5.272 × 10−3 1.00

Table 4.4: Accuracy in the l∞ norm and order of convergence of the schemes for the fourth example(H(p) = p2).


Errors and order, 5th ExampleN Monotone 2nd centered 2nd Upwind 2nd ENO64 1.328 × 10−1 - 2.105 × 10−2 – 1.129 × 10−1 - 8.577 × 10−2 -

128 1.095 × 10−1 0.27 1.111 × 10−2 0.91 3.446 × 10−2 1.69 6.995 × 10−2 0.29256 8.855 × 10−2 0.31 4.365 × 10−3 1.34 1.379 × 10−2 1.31 5.072 × 10−2 0.46512 7.043 × 10−2 0.33 2.360 × 10−3 0.88 3.772 × 10−3 1.86 1.288 × 10−2 1.97

1024 5.401 × 10−2 0.38 2.693 × 10−3 -0.19 1.870 × 10−3 1.01 8.170 × 10−3 0.66

Table 4.5: Accuracy in the l∞ norm and order of convergence of the schemes for the fifth example (H(p) =cos(p)2 + |p|).

smooth region. Therefore, in the eikonal equation case, the monotone scheme ends up not

being used in the upwind and ENO filtered schemes since these schemes have a choice on

where to interpolate, choosing to always do so on the region where the solution is smooth.

This is not however the case when the 2nd order centered scheme is used as the accurate

scheme. In this case, the filtered scheme falls back to the monotone scheme on the two grid

points adjacent to the singularity. As for the HJ equations case, the forward and backward

approximation are both always used and thus near the singularity the filtered schemes fall

back to the monotone scheme.

Order of convergence.

We first discuss the eikonal equation case. Examining Figure 4.5 and Tables 5.3, 5.4, 5.5, we

conclude that all the upwind filtered schemes have convergence rate corresponding to the

order of accuracy of the accurate scheme, except in the last example, where for the 3rd and

4th order schemes we obtain machine accuracy. This exception is explained by the fact that

in this example the solution is piecewise cubic and therefore these schemes end up being

exact (interpolating a cubic polynomial with 4 or more points yields the exact same cubic

polynomial). Obtaining the higher order convergence is in accordance with Theorem 4.2.4

since for the upwind filtered schemes the accurate scheme is always active as mentioned

above. We should point out that this higher order of convergence was already possible

to obtain using ENO schemes as is depicted in Figure 4.5 (with the sole exception of the

4th order ENO scheme in the second example, which we discuss below). Moreover, the

filtered scheme using the second centered scheme also provided second order convergence

even though as pointed above it falls into the monotone scheme near the singularity, more

precisely on the two grid points that enclose it.

In the general case of the HJ equations, the results are not as clean. In the first example,

the order of convergence remains the same with the monotone scheme still being first

order convergent. As for the second example, where the Hamiltonian is not convex, the


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

2nd upwind

2nd ENO

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−3

−2.5

−2

−1.5

−1

−0.5

0

3rd upwind

3rd ENO

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

4th upwind

4th ENO

Figure 4.3: Active stencils in the accurate schemein the last iteration for the solutions of the secondexample considered: −i means that i points to theleft were used in the interpolation.

101

102

103

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Monotone

Filtered (2nd)

Filtered (2ndUpwind)

Filtered (3rdUpwind)

Filtered (4thUpwind)

Filtered (2ndENO)

Filtered (3rdENO)

Filtered (4thENO)

101

102

103

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Monotone

Filtered (2nd)




Filtered (2ndENO)

Filtered (3rdENO)

Filtered (4thENO)

101

102

103

10−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Monotone

Filtered (2nd)




Filtered (2ndENO)

Filtered (3rdENO)

Filtered (4thENO)

Figure 4.4: Log-log plot of the errors for the one-dimensional examples of the eikonal equation.


101

102

103

10−3

10−2

10−1

100

Monotone

Filtered (2nd)


Filtered (2ndENO)

101

102

103

10−4

10−3

10−2

10−1

100

Monotone

Filtered (2nd)


Filtered (2ndENO)

Figure 4.5: Log-log plot of the errors for the one-dimensional examples of HJ equations.

monotone scheme is not even first order convergent as in all the other examples and we

see an increase in the order of convergence for both the second order upwind and ENO

filtered schemes. In general we do not expect this increase in the order of convergence of

the global accuracy since near the singularity we fall back into the monotone scheme.

Upwind vs ENO.

The ENO filtered schemes only outperformed the upwind filtered schemes in the first

example for the eikonal equation. In this example, both schemes have the same order of

convergence but with ENO schemes having a smaller asymptotic error constant, which

can be explained by the fact that the ENO schemes in this example tend to use centered

discretizations which have a smaller truncation error than the upwind discretizations. On

the other examples, the upwind filtered schemes always performed at least as good as its

ENO counterparts.

To finish the discussion, we now take a closer look at the second example for the eikonal

equation. In this, the fourth order ENO scheme does not have fourth order accuracy and

is in fact less accurate than the third order ENO scheme, which also does not have third

order accuracy. In this case, although never interpolating where the solution is singular,

the ENO scheme uses three different stencils (see Figure 4.3) which somehow seems to

prevent us to obtain the fourth order accuracy. Moreover, the second order ENO scheme

performs an interpolation where the solution is singular, although this does not affect the

order of convergence of the method (see Figure 4.3). This example illustrates the advantage

of using the upwind filtered scheme, which has a fixed stencil, over the ENO scheme,

which, while designed heuristically to choose the best stencil, may not always do so. It is


worth mentioning that the WENO schemes were introduced to improve the ENO schemes,

but these add another layer of complexity without any clear advantage over the filtered

upwind schemes.

4.3.3 Exact solutions in two dimensions

In this subsection we discuss the two dimensional examples. We consider three solutions

to the eikonal equation (4.1) with f ≡ 1, g ≡ 0 and Γ given by a circle, two points, and a

semicircle. Specifically, we have

1. Γ = (x, y) ∈ R2 | x2 + y2 = 1 ,

2. Γ =(

12, 1

2

)

,(

−12, −1

2

)

,

3. Γ = (x, y) ∈ R2 | (x2 + y2 = 1, x ≥ 0) ∨ (|y| ≤ 1, x = 0) .

We chose these examples because the corresponding solutions have varying degrees

of regularity. In the first, the solution is smooth (outside Γ). In the second we have a

singularity along the line (x, y) ∈ R2 : x = −y and therefore the solution is only piecewise

smooth outside (Γ). In the third, (outside Γ) the solution is smooth for x > 0 but only

Lipschitz continuous for x < 0. The exact solution is the distance function to the set Γ.

All computations are performed on the domain [−2, 2] × [−2, 2], which is discretized on

an N × N grid. We assume the exact solution to be known at the neighboring grid points

of Γ as discussed in subsection 4.2.3, except in the second example where we initialize

the solution where u < 0.1 in order to avoid dealing with the singularities at Γ (this is a

standard thing to do when studying the higher global accuracy of the methods [ABM+11]).

All solutions are displayed in Figure 4.6.

4.3.4 Computational results in two dimensions

In this subsection we discuss the computational results obtained in two dimensions. The

main purpose is to demonstrate that the filtered scheme achieves the higher order accuracy

in the regions where the solution is smooth. We organize the discussion in three parts:

accuracy and behaviour, order of convergence and upwind vs ENO. We obtained results

with the monotone scheme (4.5) and with the respective filtered schemes using as the

accurate scheme the second order centered, upwind and ENO schemes.


−2

−1

0

1

2

−2

−1

0

1

2

0

0.5

1

1.5

2

−2

−1

0

1

2

−2

−1

0

1

2

0

0.5

1

1.5

2

2.5

3

−2

−1

0

1

2

−2

−1

0

1

2

0

0.5

1

1.5

2

2.5

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

0.5

1

1.5

2

2.5

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Figure 4.6: Profile and contour plots of the solutions of the three examples considered in two dimensions.

Errors and order, 1st ExampleN Monotone 2nd centered 2nd Upwind 2nd ENO64 2.167 × 10−2 - 9.034 × 10−4 - 1.107 × 10−3 - 5.284 × 10−4 -128 1.126 × 10−2 0.93 2.368 × 10−4 1.91 3.030 × 10−4 1.85 1.476 × 10−4 1.82256 5.661 × 10−3 0.99 5.964 × 10−5 1.98 7.627 × 10−5 1.98 3.766 × 10−5 1.96512 2.854 × 10−3 0.99 1.516 × 10−5 1.97 1.949 × 10−5 1.96 9.682 × 10−6 1.95

1024 1.432 × 10−3 0.99 3.893 × 10−6 1.96 4.903 × 10−6 1.99 2.444 × 10−6 1.98

Table 4.6: Accuracy and order of convergence of the schemes for the first example in two dimensions in thel∞ norm.

Errors and order, 2nd ExampleN Monotone 2nd centered 2nd Upwind 2nd ENO64 5.128 × 10−2 - 1.643 × 10−2 - 1.276 × 10−2 - 1.297 × 10−2 -128 2.663 × 10−2 0.93 1.016 × 10−2 0.69 9.837 × 10−3 0.37 9.514 × 10−3 0.44256 1.326 × 10−2 1.00 5.485 × 10−3 0.88 5.121 × 10−3 0.94 4.795 × 10−3 0.98512 6.640 × 10−3 1.00 3.019 × 10−3 0.86 2.600 × 10−3 0.98 2.402 × 10−3 0.99

1024 3.324 × 10−3 1.00 1.483 × 10−3 1.02 1.425 × 10−3 0.87 1.490 × 10−3 0.69

Table 4.7: Accuracy and order of convergence of the schemes for the second example in two dimensions inthe l∞ norm.


Errors and order, 2nd ExampleN Monotone 2nd centered 2nd Upwind 2nd ENO64 4.310 × 10−1 - 4.355 × 10−2 - 7.111 × 10−2 - 3.589 × 10−2 -128 2.202 × 10−1 0.96 1.331 × 10−2 1.69 1.967 × 10−2 1.83 1.002 × 10−2 1.82256 1.088 × 10−1 1.01 2.893 × 10−3 2.19 4.844 × 10−3 2.01 2.538 × 10−3 1.97512 5.420 × 10−2 1.00 9.942 × 10−4 1.54 1.233 × 10−3 1.97 6.506 × 10−4 1.96

1024 2.706 × 10−2 1.00 2.697 × 10−4 1.88 3.149 × 10−4 1.97 1.697 × 10−4 1.94

Table 4.8: Accuracy and order of convergence of the schemes for the second example in two dimensions inthe l1 norm.

Errors and order, 3rd ExampleN Monotone 2nd centered 2nd Upwind 2nd ENO64 5.771 × 10−2 - 9.083 × 10−3 - 9.342 × 10−3 - 8.811 × 10−3 -128 3.541 × 10−2 0.70 4.833 × 10−3 0.90 5.508 × 10−3 0.75 4.566 × 10−3 0.94256 2.117 × 10−2 0.74 2.399 × 10−3 1.00 3.344 × 10−3 0.72 2.605 × 10−3 0.81512 1.238 × 10−2 0.77 1.470 × 10−3 0.70 2.523 × 10−3 0.41 1.574 × 10−3 0.72

1024 7.112 × 10−3 0.80 1.024 × 10−3 0.52 1.517 × 10−3 0.73 1.055 × 10−3 0.58

Table 4.9: Accuracy and order of convergence of the schemes for the third example in two dimensions in thel∞ norm.

Accuracy and behaviour of the filtered schemes

We begin with the results presented in Figure 4.7 and Tables 4.6, 4.7, 4.9. It is clear the

solutions computed using the filtered schemes are more accurate.

The behaviour of the filtered schemes is very much like the one obtained in the one-

dimensional examples: in first example, the monotone scheme is never used since the

solution is smooth; in the second example, it is only used near the singularity at x = −y; in

the third example, it is only used near the corners of Γ.

Order of convergence

Unlike the one dimensional case for the eikonal equation, the order of convergence of

the error in the l∞ norm can be less than the formal order of accuracy of the accurate

schemes and will depend on the smoothness of the solutions. In the first example, the

solution is smooth and we obtain second order convergence in the l∞ norm (see Figure

4.7 and Table 4.6). This was expected since the “equivalent” fast marching method was

already proven second order convergent for smooth solutions in [ABM+11]. In the second

example, we have a shock of co-dimension 1 and therefore we get first order convergence

in the l∞ norm and second order in the l1 norm (see Figures 4.7, 4.8 and Tables 4.7,

4.8). We can still recover the second order of convergence in the l∞ norm by looking


away from the singularities (see Figure 4.9). As for the third example, we do not have

shocks, but the solution is nevertheless singular due to the corners in Γ which have a

rarefaction effect much like the ones in hyperbolic conversation laws (see [QS99]). For

instance, in the region (x, y) ∈ R2 : x < 0, y > 1 all characteristics emanate from the point

(0, 1) and so the errors incurred there will propagate out and pollute the solution. Thus

the error is globally first order in both the l∞ and l1 norm. However if we restrict the

errors to the region (x, y) ∈ R2 : x2 + y2 ≥ 1, x ≥ 0.1 where the solution is smooth we

do obtain second order convergence in the l∞ norm (see Figure 4.10). Finally, in region

(x, y) ∈ R2 : |y| ≤ 0.8, x ≤ 0, all the schemes were exact up to machine precision since

they are exact on flat regions.

Upwind vs ENO

Comparing the upwind schemes to the ENO schemes, we see that we obtained similar

results with the difference being a smaller asymptotical constant. This is explained by the

fact that ENO schemes tend to use centered discretizations which have a smaller truncation

error than the upwind discretizations.

Third order upwind and ENO filtered schemes were also used, but they did not show

any advantage over the second order schemes. We did not even obtain the third order

convergence for the first example even though the solution is smooth. This is most likely

related to a result proven in [ABM+11]. There the authors show that the “equivalent” third

order fast marching method is unstable. They also provide an alternative scheme which

uses full two-dimensional stencils and that it is provably third order globally convergent

in the l∞ norm for smooth solutions. We expect that if we use that scheme as our accurate

scheme we would obtain a filtered scheme with the same order of convergence.

4.4 CONC LUS IONS

We introduce filtered schemes for HJ equations, which allow us to construct convergent,

high order accurate finite difference schemes. These schemes are extremely flexible in the

choice of accurate scheme, and so they allow for a wide range of existing discretizations

(even unstable ones) to be used, while retaining the stability and convergence proof of the

monotone schemes.

Focusing on the special, but important case of the eikonal equation, we tested the

accuracy of several discretizations on solutions of varying regularity in one and two

dimensions. In one dimension, we used filtered central differences, filtered higher order

upwinding, and filtered ENO schemes. In each case we obtained higher accuracy, even in

64 4.4. CONCLUSIONS

102

103

10−6

10−4

10−2

Monotone

Filtered (2nd)


Filtered (2ndENO)

102

103

10−4

10−2

Monotone

Filtered (2nd)


Filtered (2ndENO)

102

103

10−4

10−2

Monotone

Filtered (2nd)


Filtered (2ndENO)

Figure 4.7: Log-log plot of the errors for the two-dimensional examples in the l∞ norm.

102

103

10−6

10−4

10−2

100

Monotone

Filtered (2nd)


Filtered (2ndENO)

102

103

10−4

10−2

100

Monotone

Filtered (2nd)


Filtered (2ndENO)

102

103

10−4

10−2

100

Monotone

Filtered (2nd)


Filtered (2ndENO)

Figure 4.8: Log-log plot of the errors for the two-dimensional examples in the l1 norm.


102

103

10−4

10−2

Monotone

Filtered (2nd)


Filtered (2ndENO)

Figure 4.9: Log-log plot of the errors for thesecond example in the l∞ norm in regions

(x, y) ∈ R2 : |x + y| > 0.1

102

103

10−6

10−4

10−2

Monotone

Filtered (2nd)


Filtered (2ndENO)

Figure 4.10: Log-log plot of the errors forthe third example in the l∞ norm in regions

(x, y) ∈ R2 : x2 + y2 ≥ 1, x ≥ 0.1

.

regions where the solution was not smooth. For the eikonal equation case we were able

to prove the higher order convergence. This result, although very special to the eikonal

equation, illustrates the potential accuracy of the method.

Due to the explicit nature of the filtered upwind schemes we were able to use the simple

but effective fast sweeping method to compute solutions. In the case of filtered ENO, a

slower iterative method was used. We also gave a comparison using filtered ENO schemes,

and found an example where the error for ENO was greater than its formal accuracy.

The convergence results in two dimensions were more complicated, but more generic,

in that we expect similar results on more general HJ equations. In this case, for smooth

solutions, we obtained second order accuracy. The same order of accuracy has been

previously obtained by several authors using more complicated schemes as opposed to

the simplicity of the upwind filtered schemes. In particular, our filtered upwind schemes

in two dimensions are still explicit, thus allowing the use of the fast sweeping method to

obtain solutions.

The schemes developed here are simple to implement, and allow an unrestricted choice

of higher order discretizations to be used. While we mainly focused on a particular type

equation (HJ equations), it should be clear that the filtered schemes can be used in much

wider context, while still retaining the advantages of accuracy, stability and convergence

to the viscosity solution of the monotone schemes.

CHAPTER 5

2-HESSIAN EQUATION

5.1 INTR ODUC TI ON

In this chapter, we study numerical approximations of the fully nonlinear elliptic partial

differential equation (PDE), the 2-Hessian equation in three dimensions,

S2[u] = uxxuyy + uxxuzz + uyyuzz − u2xy − u2

xz − u2yz. (5.1)

We introduce a monotone discretization and a convergence proof to the viscosity solution

is provided. We also build a second order accurate finite difference solver which, while

unstable if a simple iteration is used, can be modified to converge in practice. Numerical

results are presented on solutions with varying regularity.

We focus on the Dirichlet problem

S2[u] = f, in Ω,

u = g, on ∂Ω,(2H)

where Ω is a rectangular (three dimensional box) domain, which is natural when treating

computationally prescribed curvature problems. (For other topologies, different boundary

conditions need to be used. For the torus, periodic boundary conditions can be used. For

the sphere, it is more complicated, but it should be possible to patch together several cubic

domains to obtain this topology.)

The operator is not elliptic, unless an additional constraint is imposed. This condition

is explained in Proposition 5.2.9 and if we assume that f > 0, it reduces to

d2u

dv2+

d2u

dw2≥ 0, for every orthogonal triplet of vectors (v, w, z).

In other words, the two dimensional Laplacian restricted to every plane is positive for

the function u. Hence the discretizations of the operator must also enforce the condition

above. This means that either we are working with a family of inequality constraints,

which makes the discretization very challenging, or that we need to find a way to encode

the constraints in either the PDE or the solver. Here, we pursue the second option.

67

68 5.2. BACKGROUND ON THE EQUATION

5.2 BACKG ROUND ON THE EQUATI ON

In this section, we present the background analysis for the k-Hessian equations, with

particular focus on the 2-Hessian equation in the three dimensional case. We follow the

review by Wang [Wan09].

The k-Hessian equation can be written as

Sk[u] = f (5.2)

where 1 ≤ k ≤ n, Sk[u] = σk(λ(D2u)), λ(D2u) = (λ1, . . . , λn) are the eigenvalues of the

Hessian matrix D2u and

σk(λ) =∑

i1<···<ik

λi1 . . . λik

is the k-th elementary symmetric polynomial. It includes the Poisson equation (k = 1)

∆u = f,

and the Monge-Ampère equation (k = n)

det D2u = f,

as particular cases.

We are interested in the Dirichlet problem

Sk[u] = f, in Ω,

u = g, on ∂Ω.(kH)

5.2.1 Admissible functions and ellipticity

When k is even, the k-Hessian equation lacks uniqueness: if u solves the k−Hessian equa-

tion, so does −u. Thus an additional condition is needed to ensure solution uniqueness.

Moreover, when studying the Poisson equation it is customary to focus on the case f ≥ 0,

which is equivalent to look for solutions that are subharmonic since as a result a maximum

principle holds. In the case of the Monge-Ampère equation, one imposes instead the

additional constraint that u is convex, which is required for the ellipticity of the equation.

In both cases, it is thus necessary to restrict the solutions to an appropriate class of func-

tions in order to ensure the equation has interesting properties. The general case of the

k-Hessian is not different.

Set

Γk = λ ∈ Rn | σj(λ) > 0, j = 1, . . . , k . (5.3)

Chapter 5. 2-Hessian equation 69

Γk is a symmetric cone, meaning that any permutation of λ is in Γk. When k = 1, Γ1

is the half space λ ∈ Rn | λ1 + . . . + λn > 0. When k = n, Γn is the positive cone Γn =

λ ∈ Rn | λj > 0, j = 1, . . . , n. The result is a restriction to subharmonic functions for k = 1

and convex functions for k = n, as mentioned above. We define as well Sn(Γk) as the set of

n × n real symmetric matrices with eigenvalue vector belonging to Γk.

Definition 5.2.1. A function u ∈ C2(Ω) is k−admissible if λ(D2u(x)) ∈ Γk for all x ∈ Ω.

Proposition 5.2.2. The function F : Ω×R×Rn×Sn → R given by F (x, r, p, M) = −σk(λ(M))+

f(x), which corresponds to the k−Hessian equation (kH), is degenerate elliptic when restricted to

Sn(Γk).

Proof. See [Wan09] for a proof.

5.2.2 Viscosity Solutions

In chapter 2, we discussed viscosity solutions. However, the k-Hessian equation is only

elliptic for k-admissible functions. Therefore, the definition of viscosity solution must be

adapted.

Definition 5.2.3. A function u ∈ USC(Ω) is a viscosity subsolution of (kH) if for every k-

admissible φ ∈ C2(Ω), whenever, u − φ has a local maximum at x ∈ Ω then

−σk

(

λ(D2φ(x)))

+ f(x) ≤ 0, if x ∈ Ω,

min(

−σk

(

λ(D2φ(x)))

+ f(x), u(x) − g(x))

≤ 0, if x ∈ ∂Ω.

Similarly, a function u ∈ LSC(Ω) is a viscosity supersolution of (kH) if for every k-admissible

φ ∈ C2(Ω), whenever, u − φ has a local minimum at x ∈ Ω then

−σk

(

λ(D2φ(x)))

+ f(x) ≥ 0, if x ∈ Ω,

max(

−σk

(

λ(D2φ(x)))

+ f(x), u(x) − g(x))

≥ 0, if x ∈ ∂Ω.

Finally, we call u a viscosity solution of (kH) if u is both a viscosity subsolution and viscosity

supersolution of (kH).

The next proposition illustrates the consistency of the definition of viscosity solution

with that of classical solution.

Proposition 5.2.4. If u is a k-admissible classical solution of (kH), then u is a viscosity solution.

Conversely, if u is a viscosity solution of (kH), f > 0 and u ∈ C2(Ω), then u is a k-admissible

classical solution.


Proof. We follow [Urb90]. We focus on the first assertion. Suppose u is a k-admissible

classical solution of (kH). Let φ ∈ C2(Ω) be a k-admissible test function and assume that

u − φ has a local maximum at x ∈ Ω. If x ∈ ∂Ω,

min(

−σk

(

λ(D2φ(x)))

+ f(x), u(x) − g(x))

≤ u(x) − g(x) = 0.

If x ∈ Ω, then ∇u(x) = ∇φ(x) and D2u(x) ≤ D2φ(x) and therefore

−σk

(

λ(D2φ(x)))

+ f(x) ≤ −σk

(

λ(D2u(x)))

+ f(x) = 0,

since (kH) is degenerate elliptic by Proposition 5.2.2 and u is an admissible classical

solution by assumption. Thus u is a viscosity subsolution. The proof that u is a viscosity

supersolution is similar and so we can conclude that u is a viscosity solution.

As for the second assertion, let x0 ∈ Ω and assume that assume that D2u(x0) /∈ Sn(Γk).

Since u ∈ C2(Ω), we have

u(x) = u(x0) + ∇uᵀ(x − x0) +1

2(x − x0)

ᵀD2u(x0)(x − x0) + O(

|x − x0|3)

Sn(Γk) is an open convex cone with vertex at the origin and contains Sn(Γn). Therefore,

since D2u(x0) /∈ Sn(Γk), there exists a unique α > 0 such that D2u(x0) + αI ∈ ∂Sn(Γk).

Consider the function φ given by

φ(x) = u(x0) + ∇uᵀ(x − x0) +1

2(x − x0)

ᵀD2u(x0)(x − x0) +1

2α |x − x0|2 .

φ is k-admissible and u − φ has a local maximum at x0. Hence, since u is a viscosity

subsolution,

−σk

(

λ(D2φ(x0) + αI))

+ f(x0) ≤ 0.

However, σk = 0 on ∂Sn(Γk) and f(x0) > 0, and so we obtain a contradiction.

The well-posedness and regularity of the equation was studied in [CNS85]. Here we

recall a well-posedness result.

Definition 5.2.5. We say that Ω ⊆ Rn is (k − 1)-convex if it satisfies

σk−1(κ) ≥ c0 > 0 on ∂Ω

for some positive constant c0 where κ = (κ1, . . . , κn−1) denote the principal curvatures of ∂Ω with

respect to its inner normal.


Theorem 5.2.6. Assume that Ω is a bounded (k − 1)-convex domain in Rn with C3,1 boundary

∂Ω, g ∈ C3,1 (∂Ω) and f ∈ C1,1(Ω) with f ≥ f0 > 0. Then there is a unique k-admissible solution

u ∈ C3,α(Ω) to the Dirichlet problem (kH) for some α ∈ (0, 1).

5.2.3 2-Hessian equation

In this chapter, we focus now on the the three-dimensional case of the 2-Hessian equation

S2[u] = f, (5.4)

where

S2[u] = σ2(λ) = λ1λ2 + λ1λ3 + λ2λ3. (5.5)

The 2-Hessian operator, S2[u], can be characterized in terms of the Hessian matrix as the

next Proposition illustrates. This alternative formula will be of particularly useful to define

a naive finite difference discretization for the 2-Hessian operator.

Proposition 5.2.7. For u ∈ C2(Ω), we can write

S2[u] = c(

D2u)

= uxxuyy + uxxuzz + uyyuzz − u2xy − u2

xz − u2yz

where c(M), the sum of the principal minors of M , is given by

c(M) =1

2

(

trace(M)2 − trace(M2))

. (5.6)

Proof. For a 3 × 3 matrix M , the characteristic polynomial is given by

det(M) − c(M)λ + trace(M)λ2 − λ3

If λ1, λ2 and λ3 are the eigenvalues of M then

c(M) = λ1λ2 + λ1λ3 + λ2λ3.

Therefore, using (5.5), we conclude that

S2[u] = c(

D2u)

= uxxuyy + uxxuzz + uyyuzz − u2xy − u2

xz − u2yz

as desired.

The linearization of c(M) defined in (5.6), is given by:

∇c(M) · N = trace(M) trace(N) − trace(MN).


We can apply it to obtain the linearization of the 2-Hessian operator, S2[u], for u ∈ C2(Ω),

∇S2[u] · ν = trace(D2u) trace(D2ν) − trace(D2uD2ν). (5.7)

Recall that a linear operator L[u] ≡ trace(A(x)D2u) is elliptic if the coefficient matrix A(x)

is positive definite.

Lemma 5.2.8. Let u ∈ C2(Ω). The linearization of the 2−Hessian operator (5.7) is elliptic if u is

2-admissible.

Proof. Without loss of generality, we choose coordinates such that D2u(x) is diagonal. We

can then rewrite the linearization of the 2-Hessian operator as

∇S2[u] · ν = trace(AD2ν)

where A = diag(λ2 + λ3, λ1 + λ3, λ1 + λ2). Hence, the linearization is elliptic if A is positive

definite, which is true if u is 2-admissible.

Remark 5.1. When the function u fails to be “strictly” 2-admissible, the linearization can be

degenerate elliptic, which affects the conditioning of the linear system (5.7). When u is not

2-admissible, the linear system can be unstable.

Our approach to build discretizations for the 2-Hessian operator relies on encoding the

2-admissibility of the solution into the PDE. In order to do so, an alternative characteri-

zation of the set Γ2, derived below, is used. Recall that by definition of Γk (see (5.3) with

k = 2), we have

Γ2 =

λ ∈ R3 | λ1 + λ2 + λ3 > 0, σ2(λ) > 0

.

Proposition 5.2.9. Let

Γ =

λ ∈ R3 | λ1 + λ2 > 0, λ1 + λ3 > 0, λ2 + λ3 > 0

. (5.8)

Then

Γ2 = Γ ∩ λ ∈ R3 | σ2(λ) > 0.

Proof. Proving the ⊇ part is straightforward. We then prove the inclusion ⊆. Suppose that

(λ1, λ2, λ3) ∈ Γ2. Without loss of generality we can assume that λ1 ≤ λ2 ≤ λ3. Thus, it is

sufficient to show that λ1 + λ2 > 0. Suppose that λ1 + λ2 ≤ 0. We consider two cases, each

leading to a contradiction.

• λ1 + λ2 = 0


We have λ1λ2 ≤ 0. Hence

σ2(λ) = λ1λ2 + λ1λ3 + λ2λ3 = λ1λ2 + (λ1 + λ2)λ3 = λ1λ2 ≤ 0,

contradicting the assumption σ2(λ) > 0.

• λ1 + λ2 < 0

Since λ1 ≤ λ2, we have λ1 < 0. Moreover

σ2(λ) > 0 ⇐⇒ λ3(λ1 + λ2) > −λ1λ2 ⇐⇒ λ3 < − λ1λ2

λ1 + λ2

and

λ1 + λ2 + λ3 > 0 ⇐⇒ λ3 > −λ1 − λ2.

From the above two inequalities we get

−λ1 − λ2 < − λ1λ2

λ1 + λ2

which we can rewrite as

λ1(λ1 + λ2) + λ22 < 0.

Now, since λ1 < 0 and λ1 + λ2 < 0, the left-end side of the inequality must be positive and

we have thus derived a contradiction.

Using differentiation it is straight forward to show that the function σ2 is nondecreasing

on the set Γ, which gives some insight to why the set of admissible functions is the set of

functions where S2 is elliptic.

The constraint σ2(λ) ≥ 0 will be enforced automatically in our schemes by taking a

nonnegative f in the PDE (2H). Therefore it is sufficient to look at the set Γ as defined in

(5.8). We will refer to this restriction as plane-subharmonic since it corresponds to u being

subharmonic on every plane.

5.3 D IS C RETIZATION AND S OLVER S

In this section, we explain why the naive finite difference method fails in general. We

introduce explicit, semi-implicit, and Newton solvers for the naive finite difference method,

which perform better by enforcing the plane-subharmonic constraint. This is similar

to the solvers used in [BFO10] for the Monge-Ampère equation. Then we introduce a

discretization which is monotone and thus provably convergent.


While the monotone discretization is less accurate, it has the advantage that it gives a

globally consistent monotone discretization of the operator, meaning that we can apply the

operator to non-admissible functions. This is useful because it circumvents the need for

special initial data, and allows for the parabolic (time-dependent) operator to be defined

on an unconstrained class of functions.

In addition, we could combine the monotone discretization with the naive finite dif-

ference discretization to obtain provably convergent, accurate filtered finite difference

schemes (see section 3.4 for more details). This approach combines the advantages of both

schemes, with little additional effort. In this work, we were mainly interested in comparing

the performance of the two schemes, so we did not implement the filtered scheme.

5.3.1 Naive finite difference scheme

We begin by introducing a naive finite difference discretization of the 2-Hessian operator.

This is done by simply using standard finite differences to discretize the operator. Denote

by D2,hu the discretized Hessian using standard finite differences on a uniform grid with

grid spacing h, i.e.,

D2,huijk =

Dxxuijk Dxyuijk Dxzuijk

Dxyuijk Dyyuijk Dyzuijk

Dxzuijk Dyzuijk Dzzuijk

,

where, e.g.,

Dxxuijk =ui+1,j,k − 2ui,j,k + ui−1,j,k

h2,

Dxyuijk =ui+1,j+1,k + ui−1,j−1,k − ui−1,j+1,k − ui+1,j−1,k

4h2.

We then get the discrete version of the 2-Hessian operator S2[u] as

SA2 [u] = c

(

D2,hu)

. (2H)A

Since we are using centered finite differences, this discretization is consistent, and it is

second order accurate if the solution is smooth (hence the superscript A) as the Lemma

below shows. However, this scheme is not monotone due to the off-diagonal terms in

the cross derivatives uxy, uxz and uyz. Therefore the Barles and Souganidis theory [BS91],

discussed in chapter 3, does not apply and no convergence proof is available.

Lemma 5.3.1. Let x ∈ Ω be a reference point on the grid and φ be a C4 function that is defined

in a neighbourhood of the grid. Then the scheme SA2 [φ] defined in (2H)A approximates (2H) with


accuracy

SA2 [φ](x) = S2[φ](x) + O(h2).

Proof. It follows from a standard Taylor series argument and the fact that all the standard

finite differences used are second order accurate.

5.3.2 Failure of the parabolic solver for the naive finite differences

In this section, we give a simple example to illustrate that the use of the naive finite

difference scheme (2H)A together with a parabolic solver fails to converge.

The parabolic solver is given by

un+1 = un − dt(−SA2 [u] + f). (5.9)

Consider the solution of (2H) in [0, 1]3, given by

u(x) =x

2

2, f(x) = 3.

The iteration is initialized with the exact solution with noise from a uniform distribution

U(−0.01, 0.01). The result after performing 25 iterations with the parabolic solver (5.9)

with time step dt = dx4 and the initial guess are illustrated in Figure 5.1. Regardless of the

time step choosen (dt = dx4/10 and dt = dx4/100 were also used), after a sufficient number

of iterations the solution behaves like in the example of Figure 5.1, until it eventually

blows up. This tells us that the instability of the parabolic solver is inherent from the

discretization rather than being the result of a poorly chosen time step. This instability is

due to the fact that there is no mechanism to pick the right solution. The discretization,

being a quadratic equation as we will see below, has two solutions: the 2-admissible

solution we are looking for and the negative of this.

5.3.3 Solvers for the naive finite difference scheme

In this section we present three different solvers for the naive finite difference scheme:

a Jacobi type solver obtained by solving the discretization for the reference variable; a

semi-implicit solver based on an identity that relates the Laplacian and the 2-Hessian

operator; a Newton solver.


00.2

0.40.6

0.81

0

0.5

1

0

0.5

1

1.5

00.2

0.40.6

0.81

0

0.5

1

0

0.5

1

1.5

Figure 5.1: Failure of the parabolic solver using the naive finite difference scheme: section z = 0.9 of theinitial guess (left) and the solution after 25 iterations (right).

5.3.3.1 Jacobi solver

The accurate discretization of (2H) leads to a quadratic equation for the reference variable

at each grid point. To see this we introduce the notation

a1 =ui+1,j,k + ui−1,j,k

2a2 =

ui,j+1,k + ui,j−1,k

2a3 =

ui,j,k+1 + ui,j,k−1

2

a4 =ui+1,j+1,k + ui−1,j−1,k

2a5 =

ui−1,j+1,k + ui+1,j−1,k

2a6 =

ui+1,j,k+1 + ui−1,j,k−1

2

a7 =ui−1,j,k+1 + ui+1,j,k−1

2a8 =

ui,j+1,k+1 + ui,j−1,k−1

2a9 =

ui,j+1,k−1 + ui,j−1,k+1

2

(5.10)

Using (2H)A, SA2 [u] = f can be rewritten as

4

h4

∑

i1<i2≤3

(ai1 − uijk)(ai2 − uijk)

= fijk +1

4h4

4∑

p=2

(a2p − a2p+1)2.

Solving for uijk and selecting the smaller root (in order to select the locally more plane-

subharmonic solution), we obtain

uijk =a1 + a2 + a3

3− 1

12

√

√

√

√8∑

i1<i2≤3

(ai1 − ai2)2 + 34∑

p=2

(a2p − a2p+1)2 + 12fijkh4. (5.11)

We can now use a Jacobi iteration to find the fixed point of (5.11). Notice that the

plane-subharmonic constraint is not enforced beyond the selection of the smaller root

in (5.11).


Remark 5.2. Formula (5.11) can be rewritten as

uijk =a1 + a2 + a3

3− h2

6

√

trace(D2,huijk)2 + 3(

fijk − S2,Ah [u]

)

.

Remark 5.3. Formula (5.11) can also be used in a Gauss-Seidel iteration, which should

converge faster than the Jacobi iteration. We chose not to implement it here since all

computational results were obtained in MATLAB, which is known to be slow with loops.

In order to prove the convergence of the above solver, it would be sufficient to show

that it is monotone, which in this case is the same as showing that the value uijk is a

nondecreasing function of the neighboring values [Obe06a]. However, this is not the case

for (5.11).

5.3.3.2 Semi-implicit solver

The next solver we discuss is a semi-implicit one, which involves solving a Laplace

equation at each iteration.

We begin with the following identity for the Laplacian in three dimensions:

|∆u| =√

(∆u)2 =√

u2xx + u2

yy + u2zz + 2uxxuyy + 2uxxuzz + 2uyyuzz.

If u solves the 2-Hessian equation, then

|∆u| =√

(∆u)2 =√

u2xx + u2

yy + u2zz + 2u2

xy + 2u2xz + 2u2

yz + 2f =√

|D2u|2 + 2f.

This leads to a semi-implicit scheme for solving the 2-Hessian equation given by

∆un+1 =√

|D2un|2 + 2f. (5.12)

Note that if u is a 2-admissible function, then ∆u ≥ 0, a condition the scheme enforces.

A good initial value for the iteration is given by the solution of

∆u0 =√

2f.

5.3.3.3 Newton solver

To solve the discretized equation SA2 [u] = f we can also use a damped Newton iteration

un+1 = un − αvn,

where 0 < α ≤ 1. The damping parameter α is chosen at each step to ensure that the

residual∥

∥

∥SA2 [un] − f

∥

∥

∥ is decreasing. (In practice we can often take α = 1, but damping is


sometimes needed.) The corrector vn solves the linear system

(

∇uSA2 [un]

)

vn = SA2 [un] − f.

To setup the above equation we need the Jacobian of the scheme, which is given by

∇uSA2 [u] =

∑

ν1,ν2∈x,y,z,ν1 6=ν2

(Dν1ν1u)Dν2ν2 − (Dν1ν2u)Dν1ν2 .

Notice that this corresponds to the discrete version of the linearization of the 2-Hessian

equation (5.7).

5.3.4 Monotone finite difference scheme

In this section we construct a monotone finite difference scheme. As we saw before, the

naive approach of simply using standard finite differences for the terms in the Hessian

matrix will not work because the cross derivative terms uxy, uxz and uyz are not monotone.

Instead the idea is to use wide stencils and a rotated coordinate system in which the

Hessian matrix is diagonal. However, this coordinate system must be found in a monotone

way. This is achieved in different steps. First, we extend the function σ2 (5.5) to be

nondecreasing in R3. This then allows us to find an equivalent expression for the 2-Hessian

operator S2[u] which can not only be discretized in a monotone manner but also encodes

the 2-admissibility constraint into the operator. Finally, we present the monotone finite

difference scheme.

We start by finding a nondecreasing extension of σ2 from Γ to R3. The reason for this is

that since we know that the eigenvalues of admissible solutions u belong to the set Γ, we

are free to redefine σ2 outside of Γ, which will prove useful later to ensure convergence.

Lemma 5.3.2. The function σ2 = f sort where sort denotes the sorting function and f is given

by

f(x, y, z) = x max(y, |x|) + x max(z, |x|) + max(y, |x|) max(z, |x|)

extends σ2 on Γ and is nondecreasing in R3.

Proof. Without loss of generality, we assume that x ≤ y ≤ z since sorting the values is

monotone. Moreover, we can rewrite f as

f(x, y, z) = max (y + x, |x| + x) max (z + x, |x| + x) − x2.

Suppose (x, y, z) ∈ Γ, then we recover σ2(x, y, z).


Next we show that σ2 is nondecreasing as a function of (x, y, z). We have two cases to

consider:

• x + y ≥ 0

Since x ≤ y ≤ z, (x, y, z) ∈ Γ and so we recover σ2 which we know to be a nondecreasing

function in Γ.

• x + y < 0

Since x ≤ y ≤ z, x < 0. We then get σ2(x, y, z) = −x2, which is increasing since x < 0.

Hence σ2 is nondecreasing.

Now, we derive an alternative formula for the 2-Hessian operator. This, together with

the extension of σ2 derived in Lemma 5.3.2 will allow us to derive an expression for the

2-Hessian operator that can be discretized in a monotone way and that encodes the 2-

admissibility constraint. The idea is to use a matrix identity, following the ideas in [FO11b]

for the Monge-Ampère equation.

Lemma 5.3.3. Let M be a 3 × 3 symmetric matrix and V be the set of all orthonormal bases of R3:

V =

(ν1, ν2, ν3) | νi ∈ R3, νi ⊥ νj if i 6= j, ‖νi‖2 = 1

.

Then

c(M) = min(ν1,ν2,ν3)∈V

σ2

(

νT1 Mν1, νT

2 Mν2, νT3 Mν3

)

. (5.13)

Proof. First note that trace(M) is invariant over conjugation OT MO by orthogonal matrices

O. Second note that trace(M2) =∑

ij m2ij ≥ ∑

i m2ii with equality when M is diagonal.

Hence, for any orthogonal matrix O, we have

trace(M)2 − trace(M2) = trace(OT MO)2 − trace(OT M2O)

= trace(OT MO)2 − trace(

(OT MO)2)

≤ trace(OT MO)2 −∑

i

(OT MO)2ii.

Therefore

2c(M) = minOT O=I,

R=OT MO

(

∑

i

rii

)2

−∑

i

r2ii

,

which can be rewritten as

c(M) = minOT O=I,

R=OT MO

σ2(diag(R)), (5.14)


where diag(R) = (r11, r22, r33) is the vector formed by the elements in the diagonal of the

matrix R and σ2 is defined by (5.5).

We can now use Lemma 5.3.3 to characterize the 2-Hessian operator of a C2 function

by expressing it in terms of second directional derivatives of u as follows:

S2[u] = min(ν1,ν2,ν3)∈V

σ2

(

∂2u

∂ν21

,∂2u

∂ν22

,∂2u

∂ν23

)

. (5.15)

The equation S2[u] = f is elliptic only on the space of 2-admissible functions (see

Proposition 5.2.2) and so the 2-admissibility of the solution needs to be treated as an

additional constraint. As mentioned before, there are two possible approaches: develop

numerical methods that enforce this constraint or absorb the constraint into the PDE

operator to produce an equivalent equation that is globally elliptic and will automatically

select the 2-admissible function. We follow the second approach by considering the

equation S2[u] = f where

S2[u] = min(ν1,ν2,ν3)∈V

σ2

(

∂2u

∂ν21

,∂2u

∂ν22

,∂2u

∂ν23

)

. (5.16)

Thus we are also interested in the Dirichlet problem

S2[u] = f, in Ω,

u = g, on ∂Ω.(2H)

We show that S2 is globally elliptic.

Proposition 5.3.4. The function F : Ω × R × Rn × Sn → R given by

F (x, r, p, M) = − min(ν1,ν2,ν3)∈V

σ2 (νᵀ

1 Mν1, νᵀ

2 Mν2, νᵀ

3 Mν3) + f(x),

which corresponds to (2H), is degenerate elliptic.

Proof. We first notice that we can rewrite F as

F (x, r, p, M) = max(ν1,ν2,ν3)∈V

−σ2 (νᵀ

1 Mν1, νᵀ

2 Mν2, νᵀ

3 Mν3) + f(x)

Let M, N ∈ Sn be such that M N and (ν1, ν2, ν3) ∈ V . Then, νᵀ

i Nνi ≤ νᵀ

i Mνi for i = 1, 2, 3

by definition of M N . Hence, since σ2 is nondecreasing in R3 by Lemma 5.3.2,

σ2 (νᵀ

1 Nν1, νᵀ

2 Nν2, νᵀ

3 Nν3) ≤ σ2 (νᵀ

1 Mν1, νᵀ

2 Mν2, νᵀ

3 Mν3)


and therefore F (x, r, p, M) ≤ F (x, r, p, N) as desired.

Unlike (2H), (2H) is globally degenerate elliptic and so no additional care is needed in

the definition of viscosity solution. We simply use the definition given in chapter 2.

Proposition 5.3.5. Suppose f > 0. Then u is a viscosity solution of (2H) if and only if u is a

viscosity solution of (2H).

Proof. The proof follows from Lemmas 5.3.6 and 5.3.7 below.

Lemma 5.3.6. Suppose f > 0. Then u is a viscosity subsolution of (2H) if and only if u is a

viscosity subsolution of (2H).

Proof. Suppose u is a viscosity subsolution of (2H). Let φ ∈ C2(Ω) be a test function and

x0 ∈ Ω be a local maximum of u − φ.

We consider first the case where x0 ∈ Ω. Then, if φ is 2-admissible, i.e., D2φ(x0) ∈ S3(Γ2),

then

S2[φ](x0) = S2[φ](x0) ≥ f(x0)

since u is a viscosity subsolution of (2H).

If φ is not a 2-admissible function, i.e., D2φ(x0) /∈ S3(Γ2), there exists a unique α > 0

such that D2φ(x0) + 2αI ∈ ∂S3(Γ2). Let φ(x) = φ(x) + α |x − x0|2. Then u − φ has a local

maximum at x0 and so

S2[φ](x0) ≥ f(x0) > 0.

Since S2[φ](x0) = 0 due to the choice of α, the above inequality gives a contradiction.

Suppose now that x0 ∈ ∂Ω. If D2φ(x0) ∈ S3(Γ2), then S2[φ](x0) = S2[φ](x0) and so

min(

−S2[φ](x0) + f(x0), u(x0) − g(x0))

= min (−S2[φ](x0) + f(x0), u(x0) − g(x0)) ≤ 0.

since u is a viscosity subsolution of (2H). Otherwise, if D2φ(x0) /∈ S3(Γ2), then, as before,

there exists a unique α > 0 such that D2φ(x0)+2αI ∈ ∂S3(Γ2). Let φ(x) = φ(x)+α |x − x0|2.

Then u − φ has a local maximum at x0 and so

min(

−S2[φ](x0) + f(x0), u(x0) − g(x0))

≤ 0

If min(

−S2[φ](x0) + f(x0), u(x0) − g(x0))

= u(x0) − g(x0), then u(x0) − g(x0) ≤ 0 and so

min(

−S2[φ](x0) + f(x0), u(x0) − g(x0))

≤ u(x0) − g(x0) ≤ 0

Otherwise, min(

−S2[φ](x0) + f(x0), u(x0) − g(x0))

= −S2[φ](x0) + f(x0), in which case

S2[φ](x0) ≥ f(x0) > 0. Then, as before, we have derived a contradiction since S2[φ](x0) = 0.


Suppose now that u is a viscosity subsolution of (2H). Let φ ∈ C2(Ω) be a 2-admissible

test function and x0 ∈ Ω be a local maximum of u − φ. Then S2[φ] = S2[φ] and so

−S2[φ](x0) + f(x0) = −S2[φ](x0) + f(x0) ≤ 0

if x ∈ Ω and

min (−S2[φ](x0) + f(x0), u(x0) − g(x0)) = min(

−S2[φ](x0) + f(x0), u(x0) − g(x0))

≤ 0.

if x ∈ ∂Ω, since φ is 2-admissible and u is a viscosity subsolution of (2H).

Lemma 5.3.7. Suppose f > 0. Then u is a viscosity supersolution of (2H) if and only if u is a

viscosity supersolution of (2H).

Proof. Suppose u is a viscosity supersolution of (2H). Let φ ∈ C2(Ω) be a test function and

x0 ∈ Ω be a local minimum of u − φ. Then, if D2φ(x0) ∈ S3(Γ2), then S2[φ](x0) = S2[φ](x0)

and so, since u is a viscosity supersolution of (2H),

−S2[φ](x0) + f(x0) = −S2[φ](x0) + f(x0) ≥ 0

if x ∈ Ω, and

max(

−S2[φ](x0) + f(x0), u(x0) − g(x0))

= max (−S2[φ](x0) + f(x0), u(x0) − g(x0)) ≥ 0.

if x ∈ ∂Ω.

Otherwise, D2φ(x0) /∈ S3(Γ2) then

S2[φ](x0) ≤ 0 < f(x0)

by definition of σ2. Thus, we can conclude that u is a viscosity supersolution of (2H).

Suppose now that u is a viscosity supersolution of (2H). Let φ ∈ C2(Ω) be a 2-admissible

test function and x0 ∈ Ω be a local minimum of u − φ. Then S2[φ](x0) = S2[φ](x0) and so,

since φ is 2-admissible and u is a viscosity supersolution of (2H),

−S2[φ](x0) + f(x0) = −S2[φ](x0) + f(x0) ≥ 0.

if x ∈ Ω and

max (−S2[φ](x0) + f(x0), u(x0) − g(x0)) = max(

−S2[φ](x0) + f(x0), u(x0) − g(x0))

≥ 0.

if x ∈ ∂Ω as desired.


We can finally present the monotone discretization of the 2-Hessian operator. Here we

are interested in solving problems on the three-dimensional box Ω = (a, b)3 and so the

computation domain is D = [a, b]3 and we take Gh to be a N × N × N uniform grid with

h = b−aN

, i.e.,

Gh =

(a + ih, a + jh, a + kh) | i, j, k ∈ 0, 1, . . . , N, h =b − a

N

.

In addition, we set GhV = Ω ∩ Gh and ∂Gh = Gh \ Gh

V . Define as well the distance of a grid

point x ∈ GhV to the boundary of the grid as

d(x, ∂Gh) = miny∈∂Gh

‖x − y‖∞ .

Notice that since we have a uniform grid d(x, ∂Gh) will be a multiple of h.

As for the second order derivatives, we approximate them using centered finite differ-

ences which leads to a spatial discretization with parameter h:

∂2u

∂ν2(xi) = Dννu(xi) + O(|ν|2 h2),

where Dνν is the finite difference operator for the second directional derivative in the

direction ν which lies on the finite difference grid and are given by

Dννu(xi) =1

|ν|2h2(u(xi + hν) + u(xi − hν) − 2u(xi)).

Since we lie on a grid, we consider only a finite number possible directions ν, which

introduces the directional discretization with parameter dθ. We denote by Gnθthe set of

orthogonal bases available on the grid for a stencil with width nθ. Setting

V1 =

ν ∈ Z3 | |νi| ≤ 1, ‖ν‖ 6= 0

and, for nθ ≥ 2,

Vnθ=

ν ∈ Z3 | |νi| ≤ nθ, ∀|t|<1 tν /∈ Vnθ−1

,

we let

Gnθ=

(ν1, ν2, ν3) ∈ V3nθ

| νi ⊥ νj if i 6= j

.

Finally, we can define the monotone scheme, with a stencil of width nθ, as

SM2 [u](x) = min

(ν1,ν2,ν3)∈GW (x)

σ2 (Dν1ν1u(x), Dν2ν2u(x), Dν3ν3u(x)) , (2H)M


where W (x) denotes the width of the stencil at x ∈ GhV and is given by

W (x) = min

nθ,d(x, ∂Gh)

h

.

We define dθ, the angular resolution, as

dθ(x) = max(w1,w2,w3)∈V

min(ν1,ν2,ν3)∈GW (x)

max

arccos

(

wT1 ν1

‖ν1‖

)

, arccos

(

wT2 ν2

‖ν2‖

)

, arccos

(

wT3 ν3

‖ν3‖

)

.

This means that near the boundary we will have a lower angular resolution since we use

narrower stencils. This can be avoided with the use of additional boundary values at the

expense of a lower order accuracy in space (because the distances to the reference point

are not the same).

We will refer to the monotone schemes with respect to the number of points in the

stencil. For instance, the monotone scheme with the stencil of length 1 (i.e., nθ = 1) has

nS + 1 = 27 points.

Remark 5.4. Given that σ2 is a symmetric function when implementing the monotone

scheme away from the boundary we do not need to look into all the triplets in Gnθ. For

instance, for nθ = 1 we only need to look for the triplets in Table 5.1.

v1 v2 v3

(1, 1, 0) (1, −1, 0) (0, 0, 1)

(1, 0, 1) (1, 0, −1) (0, 1, 0)

(1, 0, 0) (0, 1, 1) (0, 1, −1)

(1, 0, 0) (0, 1, 0) (0, 0, 1)

Table 5.1: Elements of G1 up to permutations.

nθ 1 2 3 4 5 6

nS 26 98 290 579 1155 1731

Table 5.2: nS is the number of ν directions available in the stencil, i.e., nS = #Vnθ

We now give the proof of the convergence of the monotone scheme. In order to do that,

we first need to define our scheme at the boundary, which in this case is immediate since

the grid points are aligned with the boundary. Set

F M [u](x) =

−SM2 [u](x) + f(x), if x ∈ Gh

V ,

u(x) − g(x), if x ∈ ∂Gh.(M)


Figure 5.2: Elements of V1 (solid) and elements of V2 \ V1 (dashed).

Lemma 5.3.8. The finite difference scheme F M [u], given by (M), is elliptic (and therefore mono-

tone).

Proof. From the definition, the discrete second directional derivatives Dνν are nondecreas-

ing functions of the differences between neighbouring values and reference values, uj − ui,

where uj is one of the neighbouring values of ui in the direction ν. The scheme SM2 [u]

(2H)M is a nondecreasing combination of the operators min and σ2 (the latter proved

in Lemma 5.3.2 to be nondecreasing) applied to Dνν , and so it is also a nondecreasing

function of the differences between neighbouring values and reference values. Thus

−SM2 [u](x) + f(x) is elliptic according to Definition 3.2.1. It is also clear that u − g is elliptic.

Hence, F M [u] is elliptic and its monotonicity follows from from Proposition 3.2.2.

Lemma 5.3.9. Let x0 ∈ Ω be a reference point on the grid and φ be a smooth function that is

defined in a neighborhood of the grid. Then the scheme SM2 [φ] defined in (2H)M approximates (2H)

with accuracy

SM2 [φ](x0) = S2[φ](x0) + O((nθh)2 + dθ(x0)).

Proof. From a simple Taylor series computation we have

Dννφ(x0) =∂2φ

∂ν2(x0) + O((nθh)2).


We will omit the dependence on x0 to simplify the notation.

By definition

S2[φ] = min(ν1,ν2,ν3)∈V

σ2

(

∂2φ

∂ν21

,∂2φ

∂ν22

,∂2φ

∂ν23

)

= σ2

(

∂2φ

∂v21

,∂2φ

∂v22

,∂2φ

∂v23

)

,

where the vj are orthogonal unit vectors, which may not be in the set of grid vectors G. We

know by definition of dθ that

min(ν1,ν2,ν3)∈G

max

arccos

(

vT1 ν1

‖ν1‖

)

, arccos

(

vT2 ν2

‖ν2‖

)

, arccos

(

vT3 ν3

‖ν3‖

)

≤ dθ.

Let then w ∈ G where the above min is attained. Then the angle between between each vj

and wj is less or equal than dθ and so there is dvj such that

vj + dvj =wj

‖wj‖

with ‖dvj‖ = O(dθ).

Now we consider the discretized problem

SM2 [φ] = min

(ν1,ν2,ν3)∈Gσ2 (Dν1ν1φ, Dν2ν2φ, Dν3ν3φ)

≤ σ2 (Dw1w1φ, Dw2w2φ, Dw3w3φ)

= σ2

(

∂2φ

∂w21

,∂2φ

∂w22

,∂2φ

∂w23

)

+ O((nθh)2)

= σ2

(

∂2φ

∂v21

,∂2φ

∂v22

,∂2φ

∂v23

)

+ O((nθh)2 + dθ)

= min(ν1,ν2,ν3)∈V

σ2

(

∂2φ

∂ν21

,∂2φ

∂ν22

,∂2φ

∂ν23

)

+ O((nθh)2 + dθ)

= S2[φ] + O((nθh)2 + dθ),

where we used the fact that∂2φ

∂w2j

=∂2φ

∂v2j

+ O(dθ).

In addition, since the set of grid vectors G is a subset of the set of all orthogonal vectors

V up to scaling, we find that

SM2 [φ] = min

(ν1,ν2,ν3)∈Gσ2 (Dν1ν1φ, Dν2ν2φ, Dν3ν3φ)

≥ min(ν1,ν2,ν3)∈V

σ2 (Dν1ν1φ, Dν2ν2φ, Dν3ν3φ)


= min(ν1,ν2,ν3)∈V

σ2

(

∂2φ

∂ν21

,∂2φ

∂ν22

,∂2φ

∂ν23

)

+ O((nθh)2)

= S2[φ] + O((nθh)2),

Combining the two inequalities deduced above, we conclude the proof.

Lemma 5.3.10. Assume g ∈ C(∂Ω) and f ∈ C(Ω). Then F M [u], given by (M), is consistent

with (2H) as h → 0, nθ → ∞ and nθh → 0.

Proof. As nθ → ∞, dθ → 0. Hence the consistency of F M [u] in Ω follows from the previous

Lemma.

Consider x ∈ ∂Ω, smooth φ and sequences hn → 0, yn ∈ Gh ∩ ∂Ω, zn ∈ Gh ∩ Ω such that

yn, zn → x. For sequences that approach x along the boundary we have

limn→∞,ξ→0

F M(yn, φ(yn) + ξ, φ(yn) − φ(·)) = limn→∞,ξ→0

(φ(yn) + ξ − g(yn)) = φ(x) − g(x).

For sequences that approach x from the interior, we can use the consistency and continuity

of the interior approximation to calculate

limn→∞,ξ→0

F M(zn, φ(zn) + ξ, φ(zn) − φ(·)) = −S2[φ](x) + f(x).

Combining these results yields

lim suph→0,y∈Gh→x,ξ→0

F M(y, φ(y) + ξ, φ(y) − φ(·)) = maxφ(x) − g(x), −S2[φ](x) + f(x)

= G∗(x, φ(x), ∇φ(x), D2φ(x))

and similarly for the limit inferior condition.

Lemma 5.3.11. Assume that Ω is a bounded domain. Suppose that g ∈ C(∂Ω) and f ∈ C(Ω)

with minx∈Ω f(x) = σ > 0. Then F M [u], given by (M), is stable.

Proof. F M is consistent by Lemma 5.3.10. F M is elliptic by Lemma 5.3.8. F M is Lipschitz

by construction as it involves only addition, multiplication, and computing the minimum

of operators. Then by Lemma 3.2.8, we only need to show that there exists a strict classical

subsolution and supersolution of (2H), to conclude that F M is stable.

We propose the function w(x) = A ‖x‖22 + B with

0 < A ≤ σ

24and B ≥ max

x∈∂Ω

−A ‖x‖22 + g(x)

+σ

2

as strict classical supersolution.


Clearly w is 2-admissible since A > 0. At interior points, we substitute w into the PDE

to obtain

−S2[w](x) + f(x) = −12A + f(x) ≥ −σ

2+ σ =

σ

2

due to the choice of A. At boundary points we have

max

F (x, w(x), ∇w(x), D2w(x)), w(x) − g(x)

≥ w(x) − g(x)

= A ‖x‖22 + B − g(x)

≥ maxy∈∂Ω

−A ‖y‖22 + g(y)

+σ

2−(

−A ‖x‖22 + g(x)

)

≥ σ

2

which establishes w as a strict supersolution.

Now, we propose the function w(x) = A ‖x‖22 + B as a strict classical subsolution with

A ≥ Mf

6and B ≤ min

x∈∂Ω

−A ‖x‖22 + g(x)

− Mf ,

where Mf = maxx∈Ω f(x).

Clearly w is 2-admissible since A > 0. At interior points, we substitute w into the PDE

to obtain

−S2[w](x) + f(x) = −12A + f(x) ≤ −2Mf + Mf = −Mf

due to the choice of A. At boundary points we have

min

F (x, w(x), ∇w(x), D2w(x)), w(x) − g(x)

≤ w(x) − g(x)

= A ‖x‖22 + B − g(x)

≤ miny∈∂Ω

−A ‖y‖22 + g(y)

− Mf −(

−A ‖x‖22 + g(x)

)

≤ − Mf

which establishes w as a strict subsolution.

Theorem 5.3.12. Consider the Dirichlet problem (2H) on a bounded domain Ω where g ∈ C(∂Ω)

and f ∈ C(Ω) with minx∈Ω f(x) = σ > 0. Assume that it has a strong comparison principle (see

Definition 2.3.7). Let F M be the monotone scheme (M) and Uh,nθ ∈ C(Gh) be any of its solutions.

Then

limh→0,nθ→∞,hnθ→0

uh,nθ(x) = u(x), for all x ∈ Ω


where uh,nθ is the piecewise constant extension of Uh,nθ (3.4) and u is the viscosity solution of

(2H).

Proof. F M is consistent by Lemma 5.3.10. F M is monotone by Lemma 5.3.8. F M is stable by

Lemma 5.3.11. The result then follows from Theorem 3.1.5 and the equivalence of viscosity

solutions of (2H) and (2H) proved in Proposition 5.3.5.

Remark 5.5. The assumption that a strong comparison principle holds for (2H) was dis-

cussed in section 2.3.

5.3.5 Solvers for the monotone finite difference scheme

In this section we present two solvers for the monotone finite difference scheme.

5.3.5.1 Parabolic solver

Using the monotone discretization F M [u], the simplest solver for the 2-Hessian equation is

to use the fixed point method

un+1 = un − dtF M [un] (5.17)

which corresponds to the discrete version of the parabolic equation ut = S2[u] − f using a

forward Euler step. The fixed point iteration will be a contraction in the maximum norm

provided that we choose dt small enough, as dictated by the nonlinear CFL condition

[Obe06a], which in this case means dt = O(h4) (see section 3.2 for more details). This will

make the solver very slow. However, since F M is globally degenerate elliptic, this is a

global solver, meaning that it will converge regardless of the initial guess we choose.

5.3.5.2 Newton solver

As with the standard finite difference scheme, one can also use a (damped) Newton solver.

In this case the Jacobian for the monotone discretization is obtained using Danskin’s

Theorem [Ber03] and the product rule:

∇uSM2 [u] =

−2(Dν∗

1 ν∗

1u)Dν∗

1 ν∗

1, if Dν∗

1 ν∗

1u + Dν∗

2 ν∗

2u < 0,

∑

ν1,ν2∈ν∗

1 ,ν∗

2 ,ν∗

3 ,ν1 6=ν2

(Dν1ν1u)Dν2ν2 , otherwise,

where ν∗j are the directions active in the minimum in (2H)M , with Dν∗

1 ν∗

1u ≤ Dν∗

2 ν∗

2u ≤

Dν∗

3 ν∗

3u. Unlike the previous solver, this is a local solver, meaning that we need a good

initial guess in order to have convergence.


5.4 COMP UTATIONAL R ES ULTS

In this section we summarize the results of a number of different examples using the

solvers described in the previous section. The computations are performed on a N ×N ×N

grid on the cube [0, 1]3. Unless otherwise mentioned, all solvers were initialized with an

initial guess provided by the explicit method (5.11), which we iterate until∣

∣

∣SA2 [un] − f

∣

∣

∣ <

10−1. The initial guess for the explicit method (5.11) was the exact solution with some

noise from a uniform distribution. As stopping criteria for the Newton solver we used∣

∣

∣SH2 [un] − f

∣

∣

∣ < 10−10 where H ∈ A, M. Solutions were also computed using (5.11) and

(5.12) with very similar results to the ones provided by the Newton solver being obtained.

For that reason, we chose not to display them here.

Remark 5.6. Notice that at points near the boundary of the domain, the scheme uses a

narrower stencil. Since we are interested in comparing the different wide stencils and to

avoid the lower angular resolution at the boundary in the computational results, we set

the exact solution at those points. However it is important to point out the lower angular

resolution near the boundary can be avoided by considering extra boundary points. By

doing so we can maintain the angular resolution of the wide stencil at the expense of a

lower spatial resolution since the stencil loses its symmetry.

Example 5.1 (Quadratic function). We consider the case where u is a non-convex (but

2-admissible function) given by

u(x) = x21 − 1

2x2

2 + 2x23, f(x) = 2. (5.18)

with x = (x1, x2, x3). In Table 5.3, we compare the results obtained using standard finite

differences and the monotone schemes with different stencil sizes. For this example, we

used the Newton solver for all schemes.

All methods provide machine accuracy which is expected since the standard finite

differences are exact for quadratic functions and the monotone schemes computed the

desired directional derivative.

Example 5.2 (smooth convex radial function). We consider now the case where u is given

by

u(x) = exp

(

‖x − x0‖2

2

)

, f(x) = (3 + 2‖x − x0‖2) exp(‖x − x0‖2). (5.19)

The maximum errors are given in Table 5.4. As in the previous example we used the

Newton solver for all schemes.


Errors and order, 1st ExampleN Standard Monotone (27-point) Monotone (99-point) Monotone (291-point)15 4.441 × 10−16 4.441 × 10−16 4.441 × 10−16 4.441 × 10−16

20 4.441 × 10−16 8.882 × 10−16 8.882 × 10−16 6.661 × 10−16

25 4.441 × 10−16 8.882 × 10−16 8.882 × 10−16 8.882 × 10−16

30 4.441 × 10−16 1.332 × 10−15 8.882 × 10−16 8.882 × 10−16

35 4.441 × 10−16 1.332 × 10−15 8.882 × 10−16 1.110 × 10−15

Table 5.3: Accuracy in the l∞ norm of the schemes for the first example using the Newton solver.

The standard finite differences provided second order convergence, which was expected

since the solution is smooth. The monotone schemes provided only first order convergence

(or close to it).

Errors and order, 2nd ExampleN Standard Monotone (27-point) Monotone (99-point) Monotone (291-point)15 2.393 × 10−4 - 3.472 × 10−4 - 2.167 × 10−4 - 1.302 × 10−4 -20 1.298 × 10−4 2.00 2.225 × 10−4 1.46 1.518 × 10−4 1.17 1.034 × 10−4 0.7525 8.197 × 10−5 1.97 1.650 × 10−4 1.28 1.165 × 10−4 1.13 8.552 × 10−5 0.8130 5.607 × 10−5 2.01 1.346 × 10−4 1.08 9.357 × 10−5 1.16 7.216 × 10−5 0.9035 4.091 × 10−5 1.98 1.259 × 10−4 0.42 7.809 × 10−5 1.14 6.247 × 10−5 0.91

Table 5.4: Accuracy in the l∞ norm and order of convergence of the schemes for the second example usingthe Newton solver.

Example 5.3 (smooth non-convex radial function). We consider now the case where u is

given by

u(x) = exp(

2x21 − x2

2 + 4x23

)

, f(x) = 8(

1 + 12x21 + 6x2

2 + 16x23

)

exp(

4x21 − 2x2

2 + 8x23

)

.

(5.20)

The maximum errors are given in Table 5.5. Once again the solutions were computed

with a Newton solver for all schemes.

The standard finite differences demonstrates again second order convergence. For

the monotone schemes, the error tappers off with the grid size and we only see an error

reduction by considering wider stencils. This tells us that the directional resolution

error dominates the spatial resolution error. It is important to point out that this does

not contradict our theoretical results since the only thing we proved was that we have

convergence as both h and dθ go to 0, which we observe here.


Errors and order, 3rd ExampleN Standard Monotone (27-point) Monotone (99-point) Monotone (291-point)15 3.028 × 10−4 - 3.287 × 10−2 - 1.110 × 10−2 - 5.044 × 10−3 -20 1.669 × 10−4 1.95 3.312 × 10−2 -0.02 1.211 × 10−2 -0.29 5.617 × 10−3 -0.3525 1.052 × 10−4 1.98 3.305 × 10−2 0.01 1.260 × 10−2 -0.17 5.920 × 10−3 -0.2230 7.218 × 10−5 1.99 3.311 × 10−2 -0.01 1.306 × 10−2 -0.19 6.396 × 10−3 -0.4135 5.262 × 10−5 1.99 3.302 × 10−2 0.02 1.339 × 10−2 -0.16 6.703 × 10−3 -0.29

Table 5.5: Accuracy in the l∞ norm and order of convergence of the schemes for the third example using theNewton solver.

Example 5.4 (smooth non-convex radial function). We consider another example of smooth

radial function which is non convex but 2-admissible:

u(x) = log(2 + ‖x‖2), f(x) = −4(−6 + ‖x‖2)

(2 + ‖x‖2)3. (5.21)

The maximum errors are given in Table 5.6. Once again the solutions were computed

with a Newton solver, regardless of the scheme.

As in the previous example, standard finite differences provide second order conver-

gence and only with wider stencils we see a decrease in error with the grid size. Moreover,

the monotone schemes with wider stencils also exhibit second order convergence (before

it tappers off in the case of the 99-point stencil).

Errors and order, 4th ExampleN Standard Monotone (27-point) Monotone (99-point) Monotone (291-point)15 4.723 × 10−5 - 1.664 × 10−3 - 3.882 × 10−4 - 4.909 × 10−4 -20 2.564 × 10−5 2.00 1.668 × 10−3 -0.01 1.787 × 10−4 2.54 2.500 × 10−4 2.2125 1.615 × 10−5 1.98 1.674 × 10−3 -0.01 1.007 × 10−4 2.46 1.462 × 10−4 2.3030 1.111 × 10−5 1.98 1.672 × 10−3 0.01 8.617 × 10−5 0.82 9.063 × 10−5 2.5335 8.052 × 10−6 2.02 1.670 × 10−3 0.01 9.620 × 10−5 -0.69 6.506 × 10−5 2.08

Table 5.6: Accuracy in the l∞ norm and order of convergence of the schemes for the fourth example usingthe Newton solver.

Example 5.5 (non smooth convex function). We consider now the case where u is given by

u(x) =1

2

(

(‖x − x0‖ − 0.2)+)2

, f(x) =

(

3 +1

25‖x − x0‖2− 4

5‖x − x0‖

)

1‖x−x0‖>0.2(x).

(5.22)

The maximum errors are given in Table 5.7. Due to its degenerate ellipticity, the

monotone schemes required the use of the damped Newton solver.


Despite the lack of smoothness of the solution, the Newton solver with standard finite

differences still converged. As for the monotone scheme, there was a significant increase

in the number of iterations required: the wider the stencil, the more iterations required

(around 10 times more iterations when compared to the Newton solver for the naive finite

differences in the worst cases).

For the 291-stencil, as in Example 5.3, the error tapers off, indicating that the directional

resolution error dominates the spatial error and, again, we still see the convergence as both

h and dθ go to 0.

Errors and order, 5th ExampleN Standard Monotone (27-point) Monotone (99-point) Monotone (291-point)15 7.580 × 10−4 - 2.261 × 10−3 - 7.707 × 10−4 - 5.086 × 10−4 -20 6.506 × 10−4 0.50 2.329 × 10−3 -0.10 7.235 × 10−4 0.21 1.924 × 10−4 3.1825 3.353 × 10−4 2.84 2.057 × 10−3 0.53 5.871 × 10−4 0.89 1.758 × 10−4 0.3930 3.032 × 10−4 0.53 2.156 × 10−3 -0.25 5.431 × 10−4 0.41 2.197 × 10−4 -1.1835 2.129 × 10−4 2.22 2.018 × 10−3 0.42 5.159 × 10−4 0.32 2.351 × 10−4 -0.43

Table 5.7: Accuracy in the l∞ norm and order of convergence of the schemes for the fifth example using theNewton solver.

Example 5.6 (example with blow-up). We considered as well the case

u(x) = −√

3 − ‖x‖2, f(x) = − −9 + ‖x‖2

(−3 + ‖x‖2)2. (5.23)

The maximum errors are given in Table 5.8. All solutions were computed with a Newton

solver.

Notice that f is unbounded at the boundary point (1, 1, 1) and u will be singular

at that point. Despite that the Newton solver still converged, but with a smaller rate

of convergence (approximately 0.3). It is important to observe that in the case of the

Monge-Ampère, the Newton solver failed to converge in the analogue example. This

may be because the Monge-Ampère equation is more strongly nonlinear than the 2-

Hessian equation. The better accuracy of the wider monotone schemes is explained by

the fact that the exact solution is prescribed at more grid points near the boundary of the

(computational) domain, in particular, where u is singular and f is unbounded.

Example 5.7. We consider as well the example with f ≡ 1 and zero Dirichlet boundary

conditions (g ≡ 0 ). No exact solution is known. In Figure 5.3, we illustrate some of the

surface plots of the level sets u = c of the solution with the standard finite differences


Errors and order, 6th ExampleN Standard Monotone (27-point) Monotone (99-point) Monotone (291-point)15 1.104 × 10−3 - 5.627 × 10−3 - 5.600 × 10−4 - 3.026 × 10−4 -20 1.096 × 10−3 0.02 5.224 × 10−3 0.24 4.229 × 10−4 0.92 2.628 × 10−4 0.4625 1.054 × 10−3 0.17 4.891 × 10−3 0.28 3.454 × 10−4 0.87 2.344 × 10−4 0.4930 1.007 × 10−3 0.24 4.698 × 10−3 0.21 2.921 × 10−4 0.88 2.102 × 10−4 0.5835 9.621 × 10−4 0.29 4.612 × 10−3 0.12 2.538 × 10−4 0.89 1.906 × 10−4 0.62

Table 5.8: Accuracy in the l∞ norm and order of convergence of the schemes for the sixth example using theNewton solver.

and monotone scheme where c ∈ −0.01, −0.03, −0.07. Note that the zero level set

(c = 0) is the boundary of the cube [0, 1]3 where the zero Dirichlet boundary conditions

are prescribed. The surface plots become spheres as c decreases, with c = −.01 being the

only where there is a tangible difference between the two schemes, most likely due to the

expected higher accuracy from the standard finite differences. In Figure 5.4, we plot the

curve u(t, t, t) with t ∈ [0, 1] and see that there’s a small difference between the solutions

from the standard finite differences and the monotone scheme.

Figure 5.3: Surface plots of the level sets of the solution to Example 5.7 on a 30 × 30 × 30 grid with thenaive finite differences (left) and the 27-point monotone scheme (right).

Example 5.8. We consider as well the example with f ≡ 1 and zero Dirichlet boundary


0 0.2 0.4 0.6 0.8 1−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

Newton (J)

Newton (M 27−point)

Figure 5.4: Plot of the curves t 7→ u(t, t, t) of the solution of Example 5.7 on a 30 × 30 × 30 grid.

conditions (g ≡ 0) but with a different domain Ω = Ω1 ∪ Ω2 where

Ω1 = (x, y, z) ∈ R3 : (x − 0.35)2 + (y − 0.35)2 + (z − 0.5)2 < 0.32,

Ω2 = (x, y, z) ∈ R3 : (x − 0.65)2 + (y − 0.65)2 + (z − 0.5).2 < 0.32.

No exact solution is known. To implement the boundary conditions in this case, since

the domain is not a cube, we set zero Dirichlet boundary conditions on the cube [0, 1]3

and defined f as the indicator function of Ω. In Figure 5.5, we illustrate some of the

surface plots of the level sets u = c of the solution with the standard finite differences and

monotone scheme with c ∈ 0, −0.01, −0.02, −0.03, −0.035, −0.039. In this case the zero

level set is not convex and the level sets u = c become more convex as c decreases. In this

case the difference between the standard finite differences and monotone scheme is even

smaller than in Example 5.7, as we can see in Figure 5.6, where we plot the curve u(t, t, t)

with t ∈ [0, 1].

5.5 CONC LUS IONS

The 2-Hessian equation is a fully nonlinear PDE which is elliptic provided the solutions

are restricted to a convex cone (we refer to them as plane-subharmonic functions). It is

natural to compare this equation with the Monge-Ampère PDE, which is elliptic on the

96 5.5. CONCLUSIONS

Figure 5.5: Surface plots of the level sets of the solution to Example 5.8 on a 30 × 30 × 30 grid with thenaive finite differences (left) and the 27-point monotone scheme (right).

0 0.2 0.4 0.6 0.8 1−0.04

−0.035

−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

Newton (J)

Newton (M 27−point)

Figure 5.6: Plot of the curves t 7→ u(t, t, t) of the solution of Example 5.8 on a 30 × 30 × 30 grid.


cone of convex functions and has been extensively studied numerically. In comparison,

the 2-Hessian equation is more challenging because the constraints for ellipticity are less

restrictive.

We gave two different discretizations for the 2-Hessian equation in the three-dimensional

case: a naive one obtained by simply using standard finite differences to discretize the

Hessian and a monotone discretization that takes advantage of a characterization of the

operator using a matrix inequality. Computational results were provided using exact

solutions of varying regularity and shape, from smooth to non differentiable, and from

convex to non-convex.

The naive discretization failed using a standard parabolic solver since there was no

mechanism for selecting the correct plane-subharmonic solution. Two alternative solvers

were presented, which enforced the plane-subharmonic restriction and for which experi-

mental results on a variety of solutions demonstrated that the method appears to converge.

Additionally, a Newton solver was also implemented, converging for all examples con-

sidered, even for degenerate ones or with singular right-hand sides, whenever initialized

with a good initial guess. For smooth examples, we obtained second order convergence.

The monotone discretization is stable and provably convergent but less accurate since it

requires the use of a wide stencil that introduces a directional resolution error. Numerical

examples show that the directional resolution easily dominates the spacial resolution, a

natural consequence of the three dimensional setting.

Moreover, one could have implemented filtered schemes which would provide schemes

that are provably convergent but with greater accuracy than the monotone schemes.

However, we did not implement them here, since our main goal was to compare the two

different discretizations presented and, moreover, the accurate scheme by itself proved to

be convergent for all the examples considered, even degenerate ones.

In this work, we chose the box domain since it is easier to deal with computationally as

the boundary conditions are easily implemented. Dealing with more complex boundaries

requires additional work. It is challenging to obtain higher order at the boundary while

maintaining second order directional derivatives. A natural approach would be a com-

bination of filtered schemes at the boundary and multi-scale grids [OZ15]. Unstructured

grids are another possibility, having been used successfully to solve several fully nonlinear

elliptic equations in two dimensions [Fro15].

CHAPTER 6

AFFINE CURVATURE

6.1 INTR ODUC TI ON

The planar motion of level sets by affine curvature is governed by the nonlinear partial

differential equation (PDE)

ut = Aff [u] := |∇u| (k[u])1/3 . (AC)

Here u : R2 → R, ∇u = (ux, uy) denotes the gradient of u, and k[u] denotes the curvature of

the level set of u

k[u] = div

(

∇u

|∇u|

)

=uxxu2

y − 2uxuyuxy + uyyu2x

(u2x + u2

y)3/2. (6.1)

The affine curvature PDE is closely related to the well known PDE for motion of level sets

by mean curvature

ut = ∆1u := |∇u| k[u] (MC)

studied in the seminal article [OS88]. However, the PDE (AC) exhibits instabilities not

found in the mean curvature PDE (MC), as demonstrated below. In order to resolve these

instabilities, we propose a Lipschitz regularization of the operator.

The regularized operator is also a geometric PDE, and viscosity solutions converge to

solutions of the affine curvature PDE in the limit as the regularization parameter goes to

zero [Gig06, Theorem 4.6.1]. The advantage of the regularized operator is that it allows us

to build stable, convergent explicit solvers which are not otherwise available. Moreover,

the resulting discretization can be combined into a filtered scheme that achieves the

higher accuracy of the otherwise unstable standard finite difference scheme. In addition,

the numerical solutions exhibit the affine invariance and morphology properties of the

evolution.

Our approach to the convergent discretization

In this paragraph we present an overview of our approach to build an elliptic discretization

for clarity. The details and supporting theory can be found in the sections that follow.

Since elliptic discretizations are available for ∆1u, rather than for k[u], we rewrite Aff [u] in

99

100 6.1. INTRODUCTION

terms of |∇u| and ∆1u and get

Aff [u] = A(|∇u| , ∆1u) = (|∇u|2 ∆1u)1/3, where A(p, q) =(

p2q)1/3

. (6.2)

The goal is to make use of available elliptic discretizations of ± |∇u| and ∆1u to build an

elliptic discretization of the full operator Aff [u]. However, simply inserting these operators

into the function A(p, q) is not sufficient: elliptic schemes are built by composing nonde-

creasing maps with elliptic operators, and A(p, q) fails to be nondecreasing. Furthermore,

explicit time discretizations require Lipschitz continuous operators, and A(p, q) also fails

to be Lipschitz continuous.

Since the properties of the nonlinear and singular function A(p, q) are so important, we

first study a model equation in one-dimension. Define

Aff 1D[u] := A(ux, uxx) =(

|ux|2 uxx

)1/3(6.3)

so that the Aff 1D[u] operator has the same homogeneity in first and second derivatives

as the Aff [u] operator. Like the higher dimensional PDE, the model equation exhibits the

instability of standard finite differences.

To build an elliptic scheme for the one dimensional equation, what is needed is a

nondecreasing representation of the function A(p, q), which is consistent with Aff 1D[u].

Furthermore, we need a Lipschitz continuous approximation of A(p, q), with Lipchitz

constant Kh, to build a monotone discretization of the time dependent PDE, using a time

step dt ≤ 1/Kh. Once this modified function is available, we proceed by inserting the

discretization of the two dimensional operators into the modified function, which results

in a convergent scheme.

Novelty of the work.

The scheme presented here uses the level set representation (AC). Thus it moves every

level set by the evolution, unlike thresholding methods, but requires a much smaller time

step. It is monotone, a desirable property for the convergence but also for its applications.

Moreover, it is shown numerically that it captures the affine invariance and morpholog-

ical properties. Moreover, a standard finite difference method is also considered, but

computational examples demonstrate its lack of stability.

Chapter 6. Affine curvature 101

6.2 THE AFFINE C URVATURE PDE

6.2.1 Definition of viscosity solutions

Although the theory of viscosity solutions discussed in chapter 2 is easily extended to

parabolic equations, due to their specific structure (linear dependence on ut), we present

here the specific viscosity solution definition for the affine curvature evolution. We follow

the book [Gig06]. See also the book [Cao03] for viscosity solutions for geometric evolution

equations, and numerical methods. The article [ES99] focuses on the mean curvature

equation.

First, we show the degenerate ellipticity of the affine curvature operator.

Proposition 6.2.1. The operator −Aff [u] is degenerate elliptic.

Proof. We start with the observation that using (6.1), we can write

∆1u = utt, t =(−uy, ux)

(u2x + u2

y)1/2,

where t is the (Euclidean) unit tangent. Then it is clear that −∆1 is degenerate elliptic.

Using the representation (6.2) the degenerate ellipticity of −Aff follows.

We now give the definition of viscosity solutions. Let Ω ⊆ R2 be a domain domain and

T > 0. We are interested in the Cauchy problem

ut = Aff [u] in (x, t) ∈ Ω × (0, T ),

B(x, p) := ν(x)ᵀp = 0 on (x, t) ∈ ∂Ω × (0, T ),

u(x, 0) = u0(x) in x ∈ Ω.

(6.4)

where ν is the outward unit normal of ∂Ω.

Remark 6.1. We will also be interested in the static equation Aff [u] = f . For the definition

of viscosity solution and comparison principle see the discussion in chapter 2.

Definition 6.2.2. A function u ∈ USC(Ω × [0, T ]) is called a viscosity subsolution of (6.4) if

u(x, 0) ≤ u0(x) for x ∈ Ω and for any ϕ ∈ C2(Ω × [0, T ]) such that u − ϕ has a local maximum

at (x, t) ∈ Ω × (0, T ) then

ϕt(x, t) − Aff [ϕ](x, t) ≤ 0 if x ∈ Ω,

minϕt(x, t) − Aff [ϕ](x, t), B(x, ∇φ(x, t)) ≤ 0 if ∈ ∂Ω.

102 6.2. THE AFFINE CURVATURE PDE

Similarly, u ∈ LSC(Ω × [0, T ]) is called a viscosity supersolution of (6.4) if u(x, 0) ≥ u0(x) for

x ∈ Ω and for any ϕ ∈ C2(Ω × [0, T ]) such that u − ϕ has a local minimum at (x, t) ∈ Ω × (0, T )

then

ϕt(x, t) − Aff [ϕ](x, t) ≥ 0 if x ∈ Ω,

maxϕt(x, t) − Aff [ϕ](x, t), B(x, ∇φ(x, t)) ≥ 0 if ∈ ∂Ω.

Finally, we call u a viscosity solution of (6.4) if u is both a viscosity subsolution and viscosity

supersolution.

We have the following uniqueness result.

Theorem 6.2.3 (Comparison Principle). Suppose that Ω is convex with C2 boundary ∂Ω. Let u

and v be a viscosity subsolution and supersolution of (6.4), respectively. Then u ≤ v in Ω × (0, T ).

Proof. See Theorem 3.7.1 in [Gig06].

6.2.2 Invariance properties of the PDE

We now study some properties of (AC), starting with the following Lemma.

Lemma 6.2.4. Let u ∈ C2(R2). Then the operator Aff has the following properties:

i) Rescaling: for h > 0, define v(x, y) := u(x/h, y/h)

Aff [v] = h−4/3Aff [u];

ii) Morphology: for g ∈ C1(R),

Aff [g u] = g′(u)Aff [u];

iii) Affine Invariance: for any affine map φ(x) = Ax + b,

Aff [u φ] = (det A)2/3Aff [u] φ.

Remark 6.2. For ∆1u, rescaling gives ∆1v = h−2∆1u and invariance only holds for orthogo-

nal transformations.

Proof. i) Writing, for |∇u| 6= 0, k[u] = 1|∇u|

(

tr(D2u) − ∇u ᵀD2u ∇u|∇u|2

)

, allows us to write

(Aff [u])3 = |∇u|2 tr(D2u)−∇uᵀ D2u∇u. Then for v(x, y) := u(x/h, y/h) we have ∇v = 1h∇u

and D2v = 1h2 D2u, and property i) follows.

Property ii) follows from the fact that the PDE has the structure of a level set PDE,

[Gig06], so the level sets are invariant under relabelling.


Finally, we prove iii). Setting v(x, y) = u(φ(x, y)), we have

∇v = A ∇u and D2v = AᵀD2u A. (6.5)

Moreover,

Aff [u] =(

|∇u|2 tr(D2u) − ∇uᵀ D2u ∇u)1/3

=(

uxxu2y − 2uxuyuxy + uyyu2

x

)1/3

Now, using this formula, we compute Aff [v], after which the derivatives of v are replaced

by derivatives of u using (6.5). The proof then follows from an elementary but lengthy

computation.

Theorem 6.2.5. Consider Φt : u0 7→ u(·, t) the solution map of (AC). Then Φt satisfies the

following properties:

i) Monotonicity: u ≤ v ⇒ Φt(u) ≤ Φt(v);

ii) Morphology/Relabelling: for any monotone scalar function g,

Φt(g u) = g Φt(u);

iii) Affine Invariance: for any affine map φ(x) = Ax + b,

Φt(u φ) =(

Φt(det A)2/3(u))

φ.

Remark 6.3. Note that our rescaling factor property iii) differs from the one in [Moi98],

however both formulas agree (and give unity) for special affine transformations which

have determinant 1.

Proof. We establish the three properties one by one.

i) Monotonicity: It follows easily from the fact that (AC) is an elliptic PDE and thus

satisfies a comparison principle.

ii) Morphology: Let g be a monotone scalar function and u be the solution of (AC) with

initial condition u(·, 0) = u0. We want to show that v := g u is the solution of (AC)

with initial condition v(·, 0) = g u0. Formally, it is enough to observe that

vt = g′(u)ut, Aff [v] = g′(u)Aff [u],

where the second equality follows by Lemma 6.2.4.

104 6.2. THE AFFINE CURVATURE PDE

iii) Affine Invariance: Let φ be an affine transformation with φ(x) = Ax + b. Let u be the

solution of (AC) with initial condition u(·, 0) = u0. We want to show that

v(x, y, t) := u(φ(x, y), t(det A)2/3)

is the solution of (AC) with initial condition v(·, 0) = u0 φ. We have

vt = (det A)2/3ut,

which together with Lemma 6.2.4 iii), is enough to conclude the proof.

6.2.3 An exact solution

We now give an example of an exact solution for (AC). See also [Gig06, Section 1.7.4].

Lemma 6.2.6. Given a, b > 0, let u : R2 × [0, ∞) be given by the shifted ellipse

u(x, y, t) = t +3

4(ab)2/3

(

(

x

a

)2

+(

y

b

)2)2/3

= t +3

4

(

b

ax2 +

a

by2

)2/3

.

Then u is a classical solution of (AC). Moreover, we conclude that ellipses remain ellipses of fixed

eccentricity under the motion by affine curvature.

Proof. Define

φ(s) =3(ab)2/3

4s2/3 and S(x, y) =

(

x

a

)2

+(

y

b

)2

.

We can then rewrite u as u(x, y, t) = t + (φ S)(x, y).

The proof then follows by showing that u is a solution of (AC) if and only if

1 = φ′(S) |∇S| div

(

∇S

|∇S|

)1/3

. (6.6)

Indeed, we have that

∇S(x, y) = 2(

x

a2,

y

b2

)

, |∇S| = 2ρ and ρ =

√

(

x

a2

)2

+(

y

b2

)2

.

Moreover,

div

(

∇S

|∇S|

)

=1

a2ρ− (x/a2)

2

a2ρ3+

1

b2ρ− (y/b2)

2

b2ρ3=

(y/b2)2

a2ρ3+

(x/a2)2

b2ρ3=

S(x, y)

a2b2ρ3.


Then (6.6) is equivalent to

1 = φ′(S)2ρ

(

S

a2b2ρ3

)1/3

.

To conclude the proof it is now enough to observe that φ is the solution of the equation

above together with φ(0) = 0.

6.3 NUMERIC AL METHODS FOR THE MODEL EQUATION

In this section we build convergent discretizations to the one dimensional model of our

equation defined by

Aff 1D[u] := A(ux, uxx) =(

u2x uxx

)1/3= f, x ∈ (−1, 1), (AC-1D)

as well as the parabolic PDE,

ut = Aff 1D[u] − f, for (x, t) ∈ (−1, 1) × (0, ∞),

along with initial and boundary conditions. In order to do so we need to build elliptic

discretizations for Aff 1D, which is achieved by studying the nonlinear function A(p, q). A

naive approach would suggest to simply substitute the elliptic discretizations for |ux| and

uxx into A(p, q). However, this is not possible since A(p, q) 6∈ ND(R2): dA/dp = 2/3(q/p)1/3

and so A fails to be nondecreasing when pq < 0.

Our approach is the following. First, in subsection 6.3.1, we write A(p, q) as a sum

of two nondecreasing functions in terms of |p|, − |p|, q. These terms are replaced by |uhx|,

−|uhx|, uh

xx which are elliptic and consistent and thus a consistent and elliptic discretization

for Aff 1D is built. Then, in subsection 6.3.2, we present a Lipschitz regularization of the

function A(p, q) and proceed similarly. The Lipschitz regularization is needed to build

a provably convergent explicit scheme for the parabolic PDE as discussed in section 3.2.

Finally, in subsection 6.3.3, we present the convergence results.

6.3.1 An elliptic discretization of the one dimensional operator

In the next lemma, we decompose A(p, q) into the sum of two nondecreasing functions.

Lemma 6.3.1. Define A+(p, q) = A(p+, q+) and A−(p, q) = A(p−, q−). Then A+ ∈ ND+(R2),

A− ∈ ND−(R2) and

−A(p, q) = A(p, −q) = A+(|p|, −q) + A−(−|p|, −q), for all p, q

106 6.3. NUMERICAL METHODS FOR THE MODEL EQUATION

Proof. The fact that A can be decomposed follows from checking two cases, depending on

the sign of q.

Next we establish that A+ ∈ ND+(R2) and A− ∈ ND−(R2). First it is clear that A+ ≥ 0

and that A− ≤ 0. Compute the partial derivatives,

A+p =

2

3

(

q+

p+

)1/3

≥ 0, A+q =

1

3

(

p+

q+

)2/3

≥ 0,

So A+ is nondecreasing in each variable, which is enough to show A+ ∈ ND. Similarly,

consider A−(p, q). Again taking partial derivatives, we see that A− is nondecreasing in

each variable, so A− ∈ ND.

Lemma 6.3.2. Define the finite difference operator

−Aff 1D,e[u] = A+(

∣

∣

∣uhx

∣

∣

∣

+, −uh

xx

)

+ A−(

−∣

∣

∣uhx

∣

∣

∣

−, −uh

xx

)

(AC)1D,e

where the finite difference operators involved are defined in section 3.3. Then −Aff 1D,e is elliptic

and consistent with the −Aff 1D.

Proof. The operators |uhx|+, −|uh

x|−, −uhxx are elliptic. Then, due to Lemma 6.3.1, the operator

−Aff 1D,e[u] consists of the composition of the nondecreasing functions A+ and A− with the

three preceding operators, with the first two taking the place of |p| and −|p| and the last one

taking the place of −q. Since the operators are elliptic and the functions are nondecreasing,

−Aff 1D,e[u] is elliptic. Consistency follows from the consistency of each of the schemes

used.

Remark 6.4 (Accuracy). One challenge in forming a discretization of (AC-1D) is that the

accuracy breaks down near uxx = 0. For example, if we use second order accurate

approximations for uxx, the accuracy of finite differences for the operator is only O(h2/3).

Indeed, consider the expression

A(p + hk, q + h2) = ((p + hk)2(q + h2))1/3

where k = 1 or 2. If p 6= 0, q = 0 we get

A(p + hk, q + h2) = h2/3(p + hk)2/3 = O(h2/3).

So regardless of the accuracy of the discretization of the ux term, the overall accuracy of

the scheme cannot be better than O(h2/3). In fact, a similar argument shows that the order

of accuracy is 2k/3 near p = 0, q 6= 0. Our elliptic discretization, which uses k = 1, will

have order of accuracy 2/3.


6.3.2 Lipschitz regularization of the one dimensional operator

The function A(p, q) fails to be Lipschitz continuous near the axis p = 0, q = 0. Hence

the elliptic scheme presented in the previous section is not Lipschitz, a property that is

required in order to build a monotone convergent scheme for the time dependent problem.

Thus, using the notation for the sign function sgn(q) = q/|q| for q 6= 0 and sgn(0) = 0

otherwise, we regularize A(p, q) as follows.

Definition 6.3.3. Define for K = K(δ) > 0, L = L(δ) > 0, the regularized function

Aδ(p, q) = sgn(q) min(|A(p, q)|, K|p|, L|q|). (6.7)

Naturally, we can then define the regularized PDE operator as

Aff 1D,δ[u] = Aδ(ux, uxx).

Defining an elliptic and consistent scheme for Aff 1D,δ[u] is accomplished by noticing

that the discretization (AC)1D,e generalizes when we replace A with the regularized version

Aδ in each term: just like for A(p, q) in Lemma 6.3.1, we can decompose Aδ(p, q) into the

sum of two nondecreasing functions. This is achieved in the following lemma.

Lemma 6.3.4. Define Aδ,+(p, q) = Aδ(p+, q+) and Aδ,−(p, q) = Aδ(p−, q−). Then

−Aδ(p, q) = Aδ,+(|p|, −q) + Aδ,−(−|p|, −q),

where Aδ,+ ∈ ND+(R2) and Aδ,− ∈ ND−(R2).

Proof. To prove the decomposition of Aδ we have to consider two cases: q ≥ 0 and q < 0.

Suppose first that q ≥ 0. Then Aδ,+(|p|, −q) = 0 since (−q)+ = 0. Therefore

Aδ,+(|p|, −q) + Aδ,−(−|p|, −q) = Aδ(−|p|, −q)

= − sgn(q) min (|A(p, q)|, K|p|, L|q|)= −Aδ(p, q).

Assume now that q < 0. Then Aδ,−(−|p|, −q) = 0 since (−q)− = 0. Hence

Aδ,+(|p|, −q) + Aδ,−(−|p|, −q) = Aδ(|p|, −q)

= sgn(−q) min (|A(p, q)|, K|p|, L|q|)= −Aδ(p, q).


To show that Aδ,+ is nondecreasing, we start by rewriting it as

Aδ,+(p, q) = min(

A+(p, q), Kp+, Lq+)

.

The result then follows by noticing that min, A+, Kp+, Lq+ are all nondecreasing. Similarly,

Aδ,− is also nondecreasing, which follows from rewriting it as

Aδ,−(p, q) = max(

A−(p, q), Kp−, Lq−)

.

and noticing that max, A−, Kp−, Lq− are all nondecreasing.

Ellipticity and consistency of the regularized scheme with respect to −Aff 1D,e,δ[u]

follows now easily, just like in Lemma 6.3.2. For this reason we omit the proof.

Lemma 6.3.5. For K = K(δ), L = L(δ) > 0, define the finite difference scheme

− Aff 1D,e,δ[u] = Aδ,+(

∣

∣

∣uhx

∣

∣

∣

+, −uh

xx

)

+ Aδ,−(

−∣

∣

∣uhx

∣

∣

∣

−, −uh

xx

)

. (AC)1D,e,δ

Then −Aff 1D,e,δ[u] is elliptic and consistent with −Aff 1D,δ[u].

Proving consistency of the regularized scheme with respect to −Aff 1D[u] requires extra

work as the parameters K, L need to be chosen carefully. The next theorem summarizes

the results proven in the lemmas that follow.

Theorem 6.3.6. Asssume K = h−1/3 and L = h−4/3. Let x ∈ Ω be a reference point on the grid

and φ be a smooth function that is defined in a neighborhood of the grid. Then the scheme Aff 1D,e,δ

defined by (AC)1D,e,δ is consistent with Aff 1D and has accuracy

Aff 1D,e,δ[φ](x) − Aff 1D[φ](x) = O(h2/3).

Moreover, Aff 1D,e,δ is Lipschitz continuous with constant Ch given by

Ch = h−4/3 + 2h−10/3. (6.8)

We start by showing that our regularization of A is indeed Lipschitz continuous.

Lemma 6.3.7. Suppose K√

L ≥ 1. Then

∣

∣

∣

∣

∣

d

dpAδ(p, q)

∣

∣

∣

∣

∣

≤ K,

∣

∣

∣

∣

∣

d

dqAδ(p, q)

∣

∣

∣

∣

∣

≤ L,


where the derivatives exist, and Aδ(p, q) is Lipschitz continuous. Moreover,

∣

∣

∣Aδ(p, q) − A(p, q)∣

∣

∣ ≤ max

(

4

27K2|q|, 2

3√

3L|p|)

, for all p, q.

Proof. First we determine the sets where each term in the minimum is active. We claim

that

Aδ(p, q) =

Lq, |q| ≤ L−3/2|p|,sgn(q)K|p|, |q| ≥ K3|p|,(p2q)1/3 otherwise.

Indeed, since K√

L ≥ 1, the claim follows from

|q| ≤ L−3/2|p| ⇒ K|p| ≥ L|q| and |q| ≥ K3|p| ⇒ L|q| ≥ K|p|.

and

L|q| ≤ |A(p, q)| ⇔ |q| ≤ L−3/2|p| and K|p| ≤ |A(p, q)| ⇔ K3|p| ≤ |q|.

We continue the proof by proving the derivate estimates. Computing the derivative

with respect to p gives

d

dpAδ(p, q) =

0, |q| ≤ L−3/2|p|sgn(q)K sgn(p), |q| ≥ K3|p|23(q/p)1/3 otherwise

In the third case, since |q/p| ≤ K3, the partial derivative is bounded by 23K, and so we can

conclude that |dAδ/dp| ≤ K.

Similarly, computing

d

dqAδ(p, q) =

L, q ≤ L−3/2p

0, q ≥ K3p

13(pq−1)2/3 otherwise

In the third case, since |p/q| ≤ L3/2 the value is bounded by 13L, and the global bound

holds.

Finally, we prove the estimate on the error introduced by the regularization. Since both

A and Aδ are even functions with respect to p and odd with respect to q, we can assume

without loss of generality that p and q are both positive. We have two cases to consider:

|q| ≥ K3|p| and |q| ≤ L−3/2|p| which is where A and Aδ differ.


If |q| ≥ K3|p| then∣

∣

∣Aδ(p, q) − A(p, q)∣

∣

∣ ≤ 4

27K2|q|.

Indeed, in this case, Aδ(p, q) = Kp and so

d

dp

(

A(p, q) − Aδ(p, q))

= K − 2

3

(

q

p

)1/3

= 0 ⇔ 8

27q = K3p ⇔ p =

8

27K3q

where α = 2/3. Hence

maxp

∣

∣

∣Aδ(p, q) − A(p, q)∣

∣

∣ ≤∣

∣

∣

∣

Aδ(

8

27K3q, q

)

− A(

8

27K3q, q

)∣

∣

∣

∣

=4

27K2|q|.

If |q| ≤ L−3/2|p| then∣

∣

∣Aδ(p, q) − A(p, q)∣

∣

∣ ≤ 2

3√

3L|p|.

Indeed, in this case, Aδ(p, q) = Lq and so

d

dq

(

A(p, q) − Aδ(p, q))

= L − 1

3

(

p

q

)2/3

= 0 ⇔ q = (3L)−3/2p.

Hence

maxq

∣

∣

∣Aδ(p, q) − A(p, q)∣

∣

∣ ≤∣

∣

∣Aδ(

p, (3L)−3/2p)

− A(

p, (3L)−3/2p)∣

∣

∣ =2

3√

3L|p|.

The result now follows straightforwardly.

Now, we show that Aff 1D,e,δ[u] is Lipschitz continuous.

Lemma 6.3.8. Aff 1D,e,δ[u] is Lipschitz continuous with constant

Ch =K

h+

2L

h2.

Proof. Using the derivative estimates proved in Lemma 6.3.7 and the triangle inequality,

we get

∣

∣

∣Aδ(p1, q1) − Aδ(p2, q2)∣

∣

∣ ≤∣

∣

∣Aδ(p1, q1) − Aδ(p2, q1)∣

∣

∣+∣

∣

∣Aδ(p2, q1) − Aδ(p2, q2)∣

∣

∣

≤∣

∣

∣

∣

∣

d

dpAδ(p, q1)

∣

∣

∣

∣

∣

|p1 − p2| +

∣

∣

∣

∣

∣

d

dqAδ(p2, q)

∣

∣

∣

∣

∣

|q1 − q2|

≤ K|p1 − p2| + L|q1 − q2|.

for some p, q ∈ R. Here, the schemes |uhx|± take the place of p1, p2 and have Lipschitz

constant of 1/h. On the other hand, q1, q2 are replaced by (−uhxx)± which has Lipschitz


constant 2/h2. The result then follows easily.

Finally, we study how K and L can be chosen to ensure that Aff 1D,e,δ[u] is consistent

with Aff 1D[u].

Lemma 6.3.9. Let x ∈ Ω be a reference point on the grid and φ be a C4 function that is defined in a

neighborhood of the grid. Assume K = O(h−α) and L = O(h−β) such that K√

L ≥ 1, α ∈ (0, 1)

and β ∈ (0, 2). Then the scheme Aff 1D,e,δ defined by (AC)1D,e,δ is consistent with Aff 1D and has

accuracy

Aff 1D,e,δ[φ](x) − Aff 1D[φ](x) = O(

h1−α + h2−β + hmin(2α,β/2))

.

Moreover, the optimal choice of α and β is given by α = 1/3 and β = 4/3, in which case the

accuracy is O(h2/3).

Proof. We showed in the previous lemma that

∣

∣

∣Aδ(p1, q1) − Aδ(p2, q2)∣

∣

∣ ≤ K|p1 − p2| + L|q1 − q2|.

Here,∣

∣

∣uhx

∣

∣

∣

±take the place of p1, while p2 is replaced by |ux|. On the other hand, q1 is

replaced by −uhxx, while q2 is replaced by −uxx. The result follows from the consistent of

the finite difference operators

∣

∣

∣uhx

∣

∣

∣

±= |ux| + O(h),

(

−uhxx

)±= (−uxx)± + O(h2).

Hence

Aff 1D,e,δ[φ](x) − Aff 1D,δ[φ](x) = O(

h1−α + h2−β)

.

A direct application of Lemma 6.3.7, where ux and uxx take the place of p and q

respectively, leads to the estimate

∣

∣

∣Aff 1D,δ[φ] − Aff 1D[φ]∣

∣

∣ ≤ max

(

4

27K2|uxx|, 2

3√

3L|ux|

)

.

Therefore, since K = O(h−α) and L = O(h−β)

Aff 1D,e[φ](x) − Aff 1D[φ](x) = O(

hmin(2α,β/2))

.

The accuracy of Aff 1D,e,δ then follows from the equality

Aff 1D,e,δ[φ](x) − Aff 1D[φ](x) = Aff 1D,e,δ[φ](x) − Aff 1D,δ[φ] + Aff 1D,e[φ](x) − Aff 1D[φ](x)


Finally, we observe that

max(min(1 − α, 2α)) = max(min(2 − β, β/2)) =2

3,

with the maximums being attained at α = 1/3 and β = 4/3, thus justifying the optimal

choice of α and β.

Remark 6.5. As a result of the proof, we show as well that Aff 1D,e,δ is consistent with Aff 1D,δ

with accuracy

Aff 1D,e,δ[φ](x) − Aff 1D,δ[φ](x) = O(

h1−α + h2−β)

.

Remark 6.6. With the optimal choice of α and β, the overall accuracy of Aff 1D,e,δ and Aff 1D,e

with respect to Aff 1D is the same (see Remark 6.4), meaning that no accuracy is lost due to

the regularization.

6.3.3 Convergence theorems for the one-dimensional model

Having proved the ellipticity and consistency of the schemes, the uniform convergence

follows as discussed in chapter 3.

The first convergence result is for the elliptic problem, where there is no need for the

regularized scheme.

Theorem 6.3.10. Let u(x) be the unique viscosity solution of Aff 1D[u] = f in Ω, along with

suitable boundary conditions, which we assume to satisfy a strong comparison principle (see

Definition 2.3.7). For each h > 0, let u1D,e,h be the uniformly bounded solution of Aff 1D,e[u] = f .

Then u1D,e,h → u locally uniformly, as h → 0.

Proof. Convergence for the elliptic discretization −Aff 1D,e follows from Theorem 3.1.5.

Unlike in the elliptic problem, in the parabolic problem we need to use the regularized

scheme, with the time discretization being given by a forward Euler step. This leads us to

the scheme

u(·, t + dt) = u(·, t) + dt Aff 1D,e,δ[u(·, t)]. (6.9)

Theorem 6.3.11. Let u(x, t) be the unique viscosity solution of ut = Aff 1D[u] in Ω × [0, ∞),

along with u(x, 0) = u0(x) and suitable boundary conditions. Assume as well that K = h−1/3

and L = h−4/3. For each h > 0, let u1D,e,h be the uniformly bounded solution of the monotone

time discretization (6.9) with dt ≤ 1/Ch given by (6.8). Then u1D,e,h → u locally uniformly, as

dt, h → 0.


Proof. The elliptic scheme Aff 1D,e,δ leads to the monotone time discretization (6.9) provided

dt ≤ 1/Ch where Ch is the Lipschitz constant of Aff 1D,e,δ. The convergence then follows

from Theorem 3.1.5.

Remark 6.7. It is also true that, for fixed values of K, L, there is a unique viscosity solution,

uδ, of the regularized PDE. Then, fixing K, L and using the discretization above with

dt = O(h2), the forward Euler method converges uniformly to uδ as h → 0.

6.4 NONC ONVERG ENC E OF S TANDAR D FINITE DIFFERENC ES

In this section we show that standard finite differences are unstable for both the one

dimensional and the two dimensional operators that we study. This instability can be

understood from the one dimensional model, which, if we take |ux| = 1, reduces to the

operator u1/3xx . It is certainly plausible that the singularity near uxx = 0 could lead to

instabilities. This motivates the regularization introduced in the previous section, which

replaces the singularity of the cube root with a linear function with large slope. It also

motives the convergent finite difference schemes, which have an explicit time step that

ensures the convergence of the time dependent schemes.

We begin with an example where the standard finite scheme does not converge for the

one-dimensional model. Next we consider the time dependent equation and the associated

scheme obtained with the (unregularized) elliptic scheme in space and a forward Euler

step in time. A scaling argument suggest that the choice dt = O(h4/3) should provide a

stable scheme. In fact, this scaling argument can be improved to a proof of the maximum

principle for the homogeneous equation with the same time step. However, the maximum

principle is not enough for convergence (we need the comparison principle) and we

demonstrate divergence with that time step. Using a smaller time step dt = O(h2) appears

to fix the problem. (The standard non-elliptic finite difference scheme diverges for the

example we present no matter how small the time step). The example is then generalized

to the two-dimensional operator.

6.4.1 Nonconvergence of standard finite differences in one dimension

Consider the discretization given by inserting the standard centered differences, given by

(3.7),

Aff 1D,a[u] =(

(

uhx

)2 (

uhxx

)

)1/3

.

We considered an exact solution u0(x) = sin(2πx) of (AC-1D). Then we solved the time

dependent problem, using the forward Euler time discretization, with initial data given

114 6.4. NONCONVERGENCE OF STANDARD FINITE DIFFERENCES

by the solution u(x, 0) = u0(x) We found that this solution was unstable for the standard

finite difference scheme. The results are displayed in Figure 6.1, which illustrates that the

numerical solution diverges from the exact solution. This effect persists over different grid

sizes, and over smaller time steps.

-1 -0.5 0 0.5 1

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1 -0.5 0 0.5 1

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1 -0.5 0 0.5 1

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1 -0.5 0 0.5 1

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Figure 6.1: Solution of the one-dimensional model equation using standard finite differences at timest ∈ 0, 1, 2, 5. Here dt = h2/2 on a 256-point grid.

Next we consider the discretization of the time-dependent equation

ut = Aff 1D[u] − f. (6.10)

Using a forward Euler method in time, and the elliptic method in space leads to

u(·, t + dt) = Φ1D,et (u) := u(·, t) + dt(Aff 1D,e[u(·, t)] − f). (6.11)

A scaling argument suggests dt = O(h43 ) might be stable, since the operator Aff 1D is

homogeneous to this order in h. In fact, we are able to prove the following.

Lemma 6.4.1. When f = 0 in (6.10), the solution map Φ1D,e satisfies the maximum principle

min Φ1D,et (u) ≤ Φ1D,e

t+dt(u) ≤ max Φ1D,et (u),

provided dt ≤ (h4/2)1/3.

Proof. We have

−2

h

∣

∣

∣uhx

∣

∣

∣

− ≤ −uhxx ≤ 2

h

∣

∣

∣uhx

∣

∣

∣

+.

Thus, since A+ ∈ ND+(R2) and A− ∈ ND−(R2),

0 ≤ A+(

∣

∣

∣uhx

∣

∣

∣

+, −uh

xx

)

≤ A+(

∣

∣

∣uhx

∣

∣

∣

+,

2

h

∣

∣

∣uhx

∣

∣

∣

+)

=(

2

h

)1/3∣

∣

∣uhx

∣

∣

∣

+

and

0 ≥ A+(

∣

∣

∣uhx

∣

∣

∣

−, −uh

xx

)

≥ A+(

∣

∣

∣uhx

∣

∣

∣

−, −2

h

∣

∣

∣uhx

∣

∣

∣

−)= −

(

2

h

)1/3∣

∣

∣uhx

∣

∣

∣

−

and so

−(

2

h

)1/3∣

∣

∣uhx

∣

∣

∣

− ≤ −Aff 1D,e[u] ≤(

2

h

)1/3∣

∣

∣uhx

∣

∣

∣

+


Hence

u(x) − dt(

2

h

)1/3∣

∣

∣uhx

∣

∣

∣

+ ≤ u(x) + dtAff 1D,e[u] ≤ u(x) + dt(

2

h

)1/3∣

∣

∣uhx

∣

∣

∣

−

The proof now follows due to the choice of dt and from observing that

u(x) + h∣

∣

∣uhx

∣

∣

∣

−= maxu(x + h), u(x − h), u(x)

and

u(x) − h∣

∣

∣uhx

∣

∣

∣

+= minu(x + h), u(x − h), u(x).

However, as the following example shows, the maximum principle by itself is not

enough to guarantee convergence and so the above choice for the time step is not guaran-

teed to produce a convergent scheme. In practice, the above choice for the time step leads

to a divergent but bounded scheme with nonlinear instabilities.

Example 6.1. We solved (6.10) using (6.11). We took u(x, 0) = u0 where u0(x) = Cx4/3 and

f = Aff 1D[u0]. In Figure 6.2, we plot the numerical solution obtained at different times

with dt = (h4/4)1/3 (The conservative choice of the time step is to account for the fact the

equation is not homogeneous). The exact solution grows in time. For larger times, the

solution has the same shape but with small high frequency oscillations in time. On the

other hand, choosing dt = h2/2 leads to convergence. (The data is not presented to save

space.)

-1 -0.5 0 0.5 1

-5

-4

-3

-2

-1

0

1

-1 -0.5 0 0.5 1

-5

-4

-3

-2

-1

0

1

-1 -0.5 0 0.5 1

-5

-4

-3

-2

-1

0

1

-1 -0.5 0 0.5 1

-5

-4

-3

-2

-1

0

1

Figure 6.2: Plot of the solution obtained described in Example 6.1 at time t ∈ 0, 1, 5, 20 on a 128-pointgrid.

6.4.2 Two dimensions

In this section we define and study the standard finite difference scheme for Aff [u]. We

show that the accuracy degenerates near points where the affine curvature is zero. We give

an example of a steady solution where the standard finite difference scheme breaks down.

116 6.4. NONCONVERGENCE OF STANDARD FINITE DIFFERENCES

Using (6.1), we can write

Aff [u] =(

uxxu2y − 2uxuyuxy + uyyu2

x

)1/3.

Definition 6.4.2. Let u : R2 → R. We define the scheme

Aff a[u] =(

uhxx(uh

y)2 − 2uhxuh

yuhxy + uh

yy(uhx)2)1/3

(AC)a

that approximates Aff [u].

Remark 6.8. The uhxy term is not elliptic, and consequently −Aff a is not elliptic.

Lemma 6.4.3. Let (x, y) ∈ Ω be a reference point on the grid and φ be a smooth function that is

defined in a neighborhood of the grid. Then the scheme Aff a[φ] defined by (AC)a approximates

Aff [φ] with accuracy

Aff a[φ](x, y) − Aff [φ](x, y) =

O(h2), Aff [φ](x, y) 6= 0,

O(h23 ), Aff [φ](x, y) = 0.

Proof. All the standard finite differences used are second order accurate. Therefore

φhxx

(

φhy

)2 − 2φhxφh

yφhxy + φh

yy

(

φhx

)2= φxxφ2

y − 2φxφyφxy + φyyφ2x + O(h2),

in other words, (Aff a[φ])3 = (Aff [φ])3 + O(h2). In addition, the Taylor expansion tells us

that

(r + t)1/3 = r1/3 +t

3r2/3+ O(t2)

when r 6= 0. Hence, when Aff [φ](x, y) 6= 0, it follows that Aff a[φ] = Aff [φ] + O(h2).

Otherwise, when Aff [φ](x, y) = 0, we can only conclude that Aff a[φ] = Aff [φ] + O(h23 ).

Now we present an example that shows that the scheme Aff a does not converge. We

choose a level set function and a right hand side so that the equation is a steady state (see

Example 6.6(d) for more details). Starting from initial data corresponding to the exact

solution, and evolving in time with a forward Euler step, the solution changes to order

one. Indeed, the solution does not appear to reach a steady state, even after running for

a long time. See Figure 6.3 for snapshots in time of the solution. Similar behaviour was

observed on finer grids (although it took a longer time to reach a comparable change in

the solution).


5 10 15 20 25 30

5

10

15

20

25

30

5 10 15 20 25 30

5

10

15

20

25

30

5 10 15 20 25 30

5

10

15

20

25

30

5 10 15 20 25 30

5

10

15

20

25

30

5 10 15 20 25 30

5

10

15

20

25

30

5 10 15 20 25 30

5

10

15

20

25

30

Figure 6.3: Lack of convergence when using a standard finite difference scheme: example 6.6 (d) with thestandard finite difference solver: level sets of the solution at times t = 0 (upper left), t = 15 (upper center),t = 17 (upper right), t = 20 (lower left), t = 40 (lower center) and t = 50 (lower right) with dt = h2/2 ona 32 × 32 grid.

6.5 CONVER GENT FINITE DIFFERENC E METHODS

Based on the convergence theory discussed above, and on the elliptic discretization of the

one dimensional model equation, we now build a discretization of the affine curvature

operator. This discretization is based on the median scheme for the mean curvature

operator from [Obe04]. We could also use the morphological operator [CDK95], which

results in a very similar discretization of the operator Aff [u]. We establish the accuracy of

the discretization, and show that it is elliptic. The scheme is augmented to a more accurate,

but still convergent, filtered scheme which interpolates between the standard scheme and

the elliptic scheme. We regularize the operator, which allows us to build a convergent

monotone time discretization.

6.5.1 Median for ∆1u

In [Obe04] an elliptic scheme for ∆1u(x) is presented based on taking the median of the

values of u sampled in a small approximately circular neighborhood of x. The motivation

follows from observing, using (6.1), that

∆1u = utt, t =(−uy, ux)

(u2x + u2

y)1/2,

118 6.5. CONVERGENT FINITE DIFFERENCE METHODS

where t is the (Euclidean) unit tangent. The median captures an approximation to utt,

the second tangential derivative of u, since the larger values point in the direction of the

gradient and the smaller values point in the opposite directions.

We outline the scheme here. We will define it at the reference point (x, y). Let

(x1, y1), . . . , (xnS, ynS

) be the nS neighbours and set dθ = 2πnS

. We refer to dθ as the directional

resolution. Denote the value of the solution at the point (xi, yi) by ui. Set vi = (xi, yi)−(x, y)

and (x−i, y−i) = (x, y) − vi. We choose the neighbours such that

(xi+1, yi+1) = h nθ (cos(idθ), sin(idθ)) + (ei, fi)

where |ei|, |fi| ≤ h. Thus h is the spatial resolution and nθ denotes the width of the stencil.

In fact, for nθ ≤ 5, the neighbours in the first quadrant are given by

(xi+1, yi+1) = h (bnθ cos(idθ)e, bnθ sin(idθ)e)

where i = 0, . . . , nS

4− 1, with the points on the remaining three quadrants being obtained

by π2, π and 3π

2rotations. Here bxe denotes the integer closest to x.

Figure 6.4: Neighbour points of the stencil for nθ = 3 (smaller circle) and nθ = 7 (larger circle).

nθ ns neighbours in the first quadrant1 8 (1, 0), (1, 1)

2 12 (2, 0), (2, 1), (1, 2)

3 16 (3, 0), (3, 1), (2, 2), (1, 3)

4 32 (4, 0), (4, 1), (4, 2), (3, 2), (3, 3), (2, 3), (2, 4), (1, 4)

5 40 (5, 0), (5, 1), (5, 2), (4, 2), (4, 3), (4, 4), (3, 4), (2, 4), (2, 5), (1, 5)

Table 6.1: Coordinates of the neighbours in the first quadrant of a stencil with width nθ.


Definition 6.5.1. Let u : R2 → R. Define the scheme

∆e1u(x, y) = 2

mediani=1,...,nS

ui − u(x, y)

(h nθ)2(MC)e

that approximates ∆1u.

In general, consistency of finite difference schemes follows by Taylor expansions.

However, additional care is needed when the PDE is singular, as when ∇u(x) = 0 in (MC).

We then recall here the definition of consistency for the mean curvature operator.

Definition 6.5.2. The numerical scheme F h,dθ is consistent with ∆1 if for every smooth function

φ and every (x, y) ∈ R2

limh,dθ→0

F h,dθ[φ] = ∆1φ

at (x, t) if ∇φ(x, y) 6= 0 and

λ ≤ lim infh,dθ→0

F h,dθ[φ] ≤ lim suph,dθ→0

F h,dθ[φ] ≤ Λ

at (x, t) where λ, Λ are the least and greatest eigenvalues of D2φ(x, y), otherwise.

Lemma 6.5.3. The finite difference scheme −∆e1u, given by (MC)e, is elliptic.

Proof. We can rewrite the scheme as

−∆e1u = 2

mediani=1,...,nS

(u(x, y) − ui)

(h nθ)2.

Since median is a nondecreasing function, we can immediately conclude that ∆e1u is elliptic.

Lemma 6.5.4. Let (x, y) ∈ Ω be a reference point on the grid and φ be a smooth function that

is defined in a neighborhood of the grid. Then the scheme ∆e1φ defined by (MC)e is consistent.

Further, it approximates ∆1 with accuracy

∆e1φ(x, y) = ∆1φ(x, y) + O

(

(h nθ)2 + dθ

)

,

when |∇u(x, y)| 6= 0.

Remark 6.9. Since dθ = O( 1nθ

), the “optimal” choice is h = O(n−3/2θ ), which results in

accuracy of O(h2/3). However this also requires a dense stencil. In practice, we use a fairly

narrow stencil, and combine with a filtered scheme for more accuracy.

The proof can be found in [Obe04].


6.5.2 Elliptic scheme for −Aff [u]

We now construct an elliptic scheme for −Aff [u] following the same procedure as in

subsection 6.3.1. This is accomplished by writing Aff [u] in terms of |∇u| and ∆1u as

Aff [u] = |∇u| k[u]1/3 = (|∇u|2 ∆1u)1/3 = A(|∇u| , ∆1u)

and using the elliptic schemes |∇uh|+ and −|∇uh|− presented in section 3.3 and −∆e1u

described in subsection 6.5.1.

Remark 6.10. We chose to discretize ∆1u with the median scheme (MC)e. However, other

schemes could have been used, for instance the morphological scheme in [CDK95].

Definition 6.5.5. Let u : R2 → R. We define the scheme

−Aff e[u] = A+(

∣

∣

∣∇uh∣

∣

∣

+, −∆e

1u)

+ A−(

−∣

∣

∣∇uh∣

∣

∣

−, −∆e

1u)

(AC)e

that approximates −Aff [u].

Remark 6.11. The above approach can be generalized to obtain an elliptic scheme for the

elliptic operator − |∇u| G(k[u]), where G is nondecreasing and homogeneous of order

α ≤ 1, G(tr) = tαG(r). (This PDE represents curves evolving with normal speed G(k)). In

such cases, write

|∇u| G(k[u]) = |∇u|1−α G(|∇u| k[u]) = |∇u|1−α G(∆1u).

The elliptic scheme is then given by

(

|∇uh|+)1−α

G(

(−∆e1u)+

)

+(

|∇uh|−)1−α

G(

(−∆e1u)−) .

Now we prove the ellipticity and consistency of the scheme −Aff e[u].

Lemma 6.5.6. The finite difference scheme −Aff e[u], given by (AC)e, is elliptic.

Proof. The proof is similar to Lemma 6.3.5.

Lemma 6.5.7. Let (x, y) ∈ Ω be a reference point on the grid and φ be a smooth function that is

defined in a neighborhood of the grid. Then the scheme Aff e[φ] defined by (AC)e is consistent with

Aff [φ] and has accuracy

Aff e[φ](x, y)−Aff [φ](x, y) =

O (h + (h nθ)2 + h dθ + dθ) , if Aff [φ](x, y) 6= 0,

O(

((h nθ)2 + h dθ + dθ)1/3

)

, if Aff [φ](x, y) = 0 and |∇φ| (x, y) 6= 0,

O(

h2/3)

if |∇φ| (x, y) = 0.


Proof. Suppose first that |∇φ| (x, y) 6= 0. We have

|∇φ(x, y)|2 =(

|∇φh(x, y)|±)2

+ O(h) and ∆e1φ(x, y) = ∆1φ(x, y) + O

(

dθ + (h nθ)2)

.

(See Lemma 6.5.4). Hence, when Aff [φ](x, y) 6= 0,

Aff e[φ]3 = Aff [φ]3 + O(

h + (h nθ)2 + h dθ + dθ

)

and, by the Taylor expansion like in Lemma 6.4.3, we have Aff e[φ](x, y) = Aff [φ](x, y) +

O (h + (h nθ)2 + h dθ + dθ). Otherwise, when Aff [φ](x, y) = 0, we necessarily have ∆1φ = 0

and so

Aff e[φ]3 = Aff [φ]3 + O(

(h nθ)2 + h dθ + dθ

)

.

Therefore, Aff e[φ](x, y) = Aff [φ](x, y) + O(

((h nθ)2 + h dθ + dθ)1/3

)

.

When |∇φ| (x, y) = 0,

|∇φ(x, y)|2 =(

|∇φh(x, y)|±)2

+ O(h2)

with ∆e1u being bounded. Hence, Aff e[φ] = Aff [φ] + O(h2/3).

Remark 6.12. Since dθ = O( 1nθ

), the “optimal” choice is h = O(n−3/2θ ), which results in

accuracy of O(h2/9) in the worst case. However this also requires a dense stencil. In

practice, we use a fairly narrow stencil, and combine with a filtered scheme for more

accuracy.

6.5.3 Filtered scheme for Aff [u]

Filtered schemes were discussed in section 3.4. Here, we define filtered schemes in a

slightly different form: they are a continuous linear interpolation between the accurate and

the elliptic scheme, which equals the accurate scheme when the two schemes are within ε

of each other. In order to so, we make use of the auxiliary function Sε : R2 → R, defined to

be a continuous function which for (a, b) ∈ R2 is equal to a near the diagonal and b off the

diagonal.

Definition 6.5.8. Define for ε > 0, Aε = (a, b) ∈ R2 | |a − b| < ε. Set ρ = 10ε. Define

d = dist ((a, b), Aε).

Sε(a, b) =

a if (a, b) ∈ Aε,

ρ−dρ

a + dρb if d ≤ ρ,

b otherwise.


Define the filtered scheme

Aff f [u] = Sε (Aff a[u], Aff e[u]) (AC)f

where ε = ε(h, dθ).

While theoretically, the only requirement on ε to ensure the convergence of the filtered

schemes is that ε → 0 as h, dθ → 0, in practice ε must be chosen carefully. Intuitively, it

should be large enough to permit the accurate scheme to be active where the solution

is smooth (see Proposition 3.4.9), and small enough to force the monotone scheme to be

active whenever needed for convergence (for instance, when the solution is singular) .

Remark 6.13. Proposition 3.4.9 tells us that heuristically we could choose ε =√

Acc[Aff e],

where Acc[Aff e] denotes the accuracy of Aff e (which corresponds to the choice α = βM/2

in the Proposition). In the numerical results presented here, we defined ε based on the

accuracy of the scheme away from the singularities of Aff [u].

In practice, the filtered scheme allows us to neglect the error coming from the wide

stencil, while in theory we still need to send dθ → 0 for convergence of the filtered scheme.

In our numerical examples, we obtain the accuracy of the accurate scheme in most cases.

6.5.4 Regularization and Forward Euler method

Similarly to the one dimensional model equation (see subsection 6.3.2), in order to build a

provably convergent scheme for the time dependent equation (AC) we need to regularize

the operator.

Write Aff [u] = A(|∇u| , ∆1u), where A(p, q) = (p2q)1/3, and regularize the cube root

function as before. This leads to

Aff δ[u] = Aδ(|∇u| , ∆1u),

where Aδ(p, q) = sgn(q) min(|A(p, q)|, K|p|, L|q|).Remark 6.14. The regularized operator, Aff δ[u], is still a level set operator. To see this, it is

enough to take K = L = 1/δ:

Aff δ[u] = |∇u| sgn(k[u]) min

(

|k[u]|1/3 ,|k[u]|

δ,1

δ

)

,

The operator reduces to either a multiple of the mean curvature operator, or the Eikonal

equation and otherwise we obtain Aff [u].

Similar results regarding ellipticity and consistency hold as for the one-dimensional

model, which we present without proof.


Lemma 6.5.9. For K = K(δ), L = L(δ) such that K√

L ≥ 1, define the finite difference scheme

−Aff e,δ[u] = Aδ,+(

|∇uh|+, −∆e1u)

+ Aδ,−(

−|∇uh|−, −∆e1u)

(AC)e,δ

Then, −Aff e,δ[u] is elliptic and consistent with −Aff δ[u].

As with the one-dimensional case, K and L need to be chosen appropriately in order

for −Aff e,δ[u] to be consistent with −Aff [u]. Assuming K = O(h−α), L = O(h−β) and

dθ = O(1/nθ) = O(hγ), full consistency is obtained when α ∈ (0, 1), β ∈ (0, 23) and

γ ∈ (β, 2−β2

). The optimal choices are α ∈ [1/9, 7/9], β = 4/9, γ = 2/3 and, in such case, the

accuracy is O(h2/9). The proof is similar to Lemma 6.3.9. As in the one dimensional model,

no accuracy is lost with the regularization (see Remark 6.12).

The Lipschitz constant of Aff e,δ[u] is given by

Ch =K

h+

2L

(h nθ)2, (6.12)

with the proof similar to Lemma 6.3.8. In practice, we will choose K = cKh−1/9 and

L = cLh−4/9, which leads to

Ch = cKh−10/9 + 2cL

nθ2h−22/9.

We can finally define and prove the convergence of the discretizations for the time

dependent equation (AC). The time derivative is discretized with an explicit forward

Euler step, while Aff [u] is discretized using either the regularized elliptic scheme Aff e,δ

or the regularized filtered scheme Aff f,δ[u] (this is easily defined by replacing the elliptic

scheme in Aff f by its regularized version). This leads to the monotone (resp. filtered) time

discretization with solution map given by

u(·, t + dt) = u(·, t) + dtAff e,δ[u(·, t)],(

resp. = u(·, t) + dtAff f,δ[u(·, t))

. (6.13)

6.5.5 Convergence Theorems

Having proved the ellipticity and consistency of the schemes, the uniform convergence

follows as discussed in chapter 3, provided there exists unique viscosity solutions to the

PDEs (AC) and Aff [u] = f along with the homogeneous Neumman boundary conditions,

which is assured by the theory in [Gig06], as explained above.

Existence of stable solutions to the schemes is also required. Here, we will make this an

assumption of our result for readability. In rigour, it follows from adding a small multiple


of u to the scheme which makes it proper (see the discussion in section 3.2, in particular

Remark 3.7).

There are two convergence results. The first is for the elliptic problem, where there is

no need for regularization. The second is for the parabolic problem, where we need to

regularize and use an explicit Euler time step.

Theorem 6.5.10. Let Ω be a bounded convex domain with a C2 boundary and u denote the unique

viscosity solution of Aff [u] = f in Ω, along with homogeneous Neumann boundary conditions

on ∂Ω. For each ε = ε(h, dθ) > 0, let ue,ε, (resp. uf,ε) be the uniformly bounded solution of

Aff e[u] = f , (resp. Aff f [u] = f ). Then

ue,ε → u and uf,ε → u, locally uniformly, as ε → 0.

Proof. The assumptions on Ω guarantee that the PDE satisfies a comparison principle.

Convergence for the elliptic discretization −Aff e then follows from Theorem 3.1.5. As for

the filtered schemes, it follows from Theorem 3.4.3.

Theorem 6.5.11. Let Ω be a bounded convex domain with a C2 boundary. Assume that u(x, t)

is the unique viscosity solution of ut = Aff [u] in Ω × [0, ∞), along with u(x, 0) = u0(x) and

homogeneous Neumman boundary conditions. Assume as well that K and L are picked so that

Aff e,δ is consistent with Aff . For each ε = ε(h, dθ) > 0, let ue,ε (resp. uf,ε) be the uniformly

bounded solution of the monotone (resp. filtered) time discretization (6.13) with dt ≤ 1/Ch where

Ch is the Lipschitz constant of Aff e,δ[u] (6.12). Then

ue,ε → u and uf,ε → u

locally uniformly, as dt, ε → 0.

Proof. The assumptions on Ω guarantee that the PDE satisfies a comparison principle. The

elliptic scheme leads to the monotone time discretization (6.13) provided dt ≤ 1/Ch where

Ch is the Lipschitz constant of Aff e,δ[u] (6.12). The convergence then follows from Theorem

3.1.5. For the time discretization of the filtered scheme, the convergence of filtered schemes

follows from Theorem 3.4.3.

Remark 6.15. Just like in the one-dimensional model, for fixed values of K, L, there is a

unique viscosity solution, uδ, of the regularized PDE. Then, fixing K, L and using the

discretization above with dt = O((h nθ)2), the forward Euler method converges uniformly

to uδ as h → 0.


6.6 NUMERIC AL RES ULTS

In this section, we start with a simple example to compare the affine invariant curvature

motion and the regularized model. We present different examples on the evolution of

a single curve in subsection 6.6.1 under the affine curvature motion and compare it to

the mean curvature motion. We test the accuracy of the schemes by computing solutions

to the time-independent PDE in subsection 6.6.2 and to the time dependent equation

in subsection 6.6.3. We mostly considered stationary problems because it was easier to

generate exact solutions by applying the operator Aff to a given function u, and including

f = Aff [u] as a source function. For the time dependent problem, we took advantage of

the exact solution from Lemma 6.2.6 to compare the accuracy of the solutions after a short

time, T = 0.1. We test as well in subsection 6.6.4 if the schemes satisfy numerically the

morphology and affine invariance properties that (AC) satisfies (see Theorem 6.2.5).

Definition 6.6.1 (Parameters for the discretization). We used the regularized schemes with

K = 20h−1/9 and L = 20h−4/9. For the elliptic discretization, we used two different stencils:

the narrow elliptic scheme used nθ = 3 and the wider elliptic scheme used nθ = 7. The filtered

discretization used the wider elliptic scheme with

ε(h, dθ) =√

h + dθ/10.

For the forward Euler method with the elliptic and filtered schemes, we used a constant time step

of dt = 1/Ch where Ch is given by (6.12). For the forward Euler method with the standard finite

difference scheme, we used the same time step as the filtered scheme, except when computing the

steady state solutions in Example 6.6 where we used dt = h2/2 (this choice of time step proved

to be enough in practice). The stopping criteria was simply that the l∞ norm of the residual∣

∣

∣Aff f,δ[un] − f∣

∣

∣ (and similarly for Aff e,δ, Aff a) was below tol = 10−5.

Example 6.2. [Comparing the regularized to the unregularized operators] We compare

the evolution of an ellipse by the affine invariant curvature motion and by the regularized

model. The ellipse should remain an ellipse of fixed eccentricity. The results were obtained

by numerically solving (AC) and ut = Aff δ[u] respectively, with the initial condition

u0(x, y) = min

(

x

2

)2

+ y2 − 1, 1

and homogeneous Neumann boundary conditions. We took [−4, 4]2 as the computational

domain on a 128 × 128 grid. The narrow elliptic scheme was used for the spatial discretiza-

tion. The difference between the level sets of the solution of the two equations is visually

126 6.6. NUMERICAL RESULTS

indistinguishable. Measured in the l∞ norm ranges from 10−6 for early times steps and

10−7 for the later time steps. The level sets of the numerical solution obtained by solving

the regularized PDE are depicted in Figure 6.5.

Remark 6.16. While the regularized scheme with the more stringent time step is needed for

the convergence proof, in practise we found, as reported in the previous example, using

the (unregularized) elliptic discretization with the time step dt = h2/2 gave nearly identical

results to within two or more significant digits. For the remaining examples, we present

results using the regularized schemes (with K = 20h−1/9 and L = 20h−4/9).

6.6.1 Numerical examples showing curve evolution

In this section we present some numerical examples illustrating the geometric properties

of the PDEs.

Example 6.3 (Ellipse). We compare the evolution of an ellipse by the affine invariant

curvature motion and by mean curvature. For the former, the ellipse should remain an

ellipse of fixed eccentricity. For the latter, the ellipse asymptotically approaches a circle

instead. The results were obtained by numerically solving (AC) and (MC), respectively,

with the initial condition

u0(x, y) = min

(

x

2

)2

+ y2 − 1, 1

and homogeneous Neumann boundary conditions. We took [−4, 4]2 as the computational

domain on a 128 × 128 grid. As for the scheme used, we chose the narrow elliptic schemes

for both. In Figure 6.5, we plot the zero level sets obtained at t ∈ 0, 0.1, 0.3, 0.5, 0.7, 0.9.

Example 6.4 (Diamond). We also compute the solution of (AC) with initial condition

(a) u0(x, y) = min |x| + |y| − 1, 1 , (b) u0(x, y) = min |x| + 2|y| − 1, 1

and homogeneous Neumann boundary conditions. The exact solutions are not known.

However, we do know that smooth convex curves evolving under affine curvature con-

verge to ellipses until collapsing to a point and that is the behaviour we observed here (see

Figure 6.5). We took [−2, 2]2 as the computational domain on a 128 × 128 grid. As for the

scheme, we chose again the narrow elliptic scheme. In Figure 6.5, we plot the zero level

sets of the numerical solution from time t = 0 to t = 0.5 in increments of 0.1 for example

(a) and from t = 0 to t = 0.3 in increments of 0.05 for example (b).


-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 6.5: (Top:) Evolution of an ellipse by (left) affine curvature, (right) mean curvature. (Bottom:)Evolution by affine curvature of (left) a diamond and (right) a flatter diamond.

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Figure 6.6: Evolution of a fan-shape like curve under affine curvature motion (top) and mean curvature(bottom) at time t ∈ 0, 0.05, 0.1, .2 (see Example 6.5 for more details).

Example 6.5 (Fan-shape curve). We also compute the solution of (AC) and (MC) with

initial condition

u0(x, y) = min

c(1)+ (x, y), c

(1)− (x, y), c

(2)+ (x, y), c

(2)− (x, y), 1

,

where

c(1)± (x, y) =

(

x ± 1

2

)2

+ 5(

y ± 1

4

)2

− 1

2and c

(2)± (x, y) = 5

(

x ± 1

4

)2

+(

y ∓ 1

4

)2

− 1

2

and homogeneous Neumann boundary conditions. The exact solution is not known. As in

the previous example, we took [−2, 2]2 as the computational domain on a 128 × 128 grid


together with the narrow elliptic scheme.

Under the both curvature motions, the curve should initially become convex. At later

times, under the affine curvature motion, the curve should evolve to an ellipse as opposed

to a circle in the mean curvature case. In Figure 6.6, we plot the zero level set of the

numerical solutions at time t ∈ 0, 0.05, 0.1, .2 and observe the exact behaviour described.

6.6.2 Accuracy of stationary solutions

We test the accuracy for the following Dirichlet problem

Aff [u] = f, in Ω,

u = g, on ∂Ω.

Solutions are obtained by computing the steady state solution of ut = Aff [u] − f , with

u(·, t) = g on ∂Ω and u(x, 0) = u0(x). We set dt = 1/Ch where Ch is given by (6.12) for

the elliptic and filtered schemes and dt = h2/2 for the standard finite difference scheme

scheme. These examples also demonstrate stability of the time dependent problem for

elliptic and filtered scheme, as well as convergence to the steady solution, since we used

the time dependent problem to obtain the solution.

Remark 6.17. The unregularized schemes were also used. For these we set dt = h2/2 for

all examples, except for the filtered scheme in example (d) where we set dt = h2/8. The

results obtained were virtually the same.

We set u0(x) to be the exact solution in a layer of seven grid points adjacent to the

boundary. As a result, each discretization is initialized at the same set of grid points and

therefore we can make a fair comparison of their accuracy. On the coarsest grid, we set

u0(x) = 0 on the interior grid points. To speed up calculations, on finer grids we set u0(x)

to be interpolated solutions from the coarser grids at interior grid points.

Example 6.6. We consider the following exact solutions

(a) u(x, y) = x2 + y2, f(x) = 2(

x2 + y2)

13 ,

(b) u(x, y) = ex2+y2

, f(x, y) = 2(

e3(x2+y2)(x2 + y2))

13 ,

(c) u(x, y) =(

x2 + y2)

13 , f(x) =

4

3,

(d) u(x, y) =sin(2πx) sin(2πy)

4, f(x) =

π43

2(− (2 + cos(4πx) + cos(4πy)) sin(2πx) sin(2πy))

13 ,


with Ω = [−1, 1]2. The solutions in (a),(b) and (d) are smooth, but the functions f are only

Holder continuous, C0,2/3, with singularities at the origin for (a) and (b), and at several

points in example (d). The solution in (c) is only C0,2/3 with a singularity at the origin, but

in this case the function f is constant.

The results are presented in Table 6.2. In Example 6.6(a) the solution is a quadratic

polynomial. The accurate scheme Aff a gives essentially machine precision, which is not

surprising, since this scheme is second order accurate. The filtered scheme Aff f gives

nearly the same accuracy (with a small discrepancy which could be eliminated by tuning

the parameter). On the other hand, both the narrow and wider elliptic scheme are much

less accurate. In contrast, in Example 6.6(b) the solution is smooth but not quadratic. In this

case we see that the accurate scheme appears to be converging to O(h). The elliptic schemes

are less accurate and the filtered scheme is in between. In Example 6.6(c) the solution is

only Holder continuous. In this case, the elliptic schemes have almost constant error near

0.01 over the range of parameters used. Despite the singular solution, the accurate scheme

gives accuracy O(h), and the filtered scheme does just as well. Example 6.6(d) shows

that the standard finite difference scheme does not converge as discussed in section 6.4

(see Figure 6.3). The narrow elliptic scheme has almost constant error 0.03 and the wider

elliptic scheme has error 0.1 for the smallest grid, decreasing by a factor of two as the grid

is refined. The filtered scheme has the best accuracy, achieving an error of 0.001 at the

finest grid.

When comparing the different schemes, we have to account for the width of the stencil

since for the elliptic schemes the wider schemes also have a larger spatial discretization

error. In general, the accuracy improved with the use of the wider stencil. Moreover,

the filtered scheme performed as expected by providing better accuracy than the elliptic

scheme and almost as good accuracy as the accurate scheme. The final example shows

that the standard finite difference scheme may not converge. This may be due to the fact

that, despite the solution being smooth, there were multiple points where f was singular.

Geometrically, this solution has several points where curves shrunk to zero.

We also considered for Example 6.6 the elliptic schemes with nθ = 1 and 2. These

provided poor accuracy with the directional resolution error easily dominating the spatial

resolution error. The errors do not decrease to zero as we decrease the grid size. This

property is consistent with the theoretical results since convergence is only proved as both

h, dθ → 0, which is indeed observed in the numerical results. On the other hand, using the

narrow and wider schemes, the accuracy of the elliptic scheme was good enough for the

filtered scheme to give accuracy comparable to the accurate scheme in many examples. In

principle we still need to send dθ → 0, but in practice, the rate at which it needs to go to


zero is much slower than h when the filtered scheme is used.

Finally, we point that the choice of ε is not an easy one and it is hard to pick an ε that

yields optimal results in each example presented. By choosing ε larger, it is possible to

achieve with the filtered schemes the accuracy of the standard schemes in Examples 6.6

(a), (b), (c). However, such choice is too permissive for Example 6.6 (d). As pointed out in

subsection 6.5.3, ε needs to be chosen small enough in order for the monotone scheme to be

active to ensure convergence. (This is comparable to the CFL condition in time dependent

equations: methods are convergent as dt, h → 0, with dt satisfying the CFL condition.)

Errors and order, Example 6.6 (a)

N Narrow Elliptic (nθ = 3) Wide Elliptic (nθ = 7) Standard Filter32 3.565 × 10−2 3.978 × 10−2 2.922 × 10−7 5.104 × 10−7

64 2.555 × 10−2 2.696 × 10−2 3.019 × 10−7 2.827 × 10−7

128 1.623 × 10−2 1.678 × 10−2 1.982 × 10−7 1.984 × 10−7

256 1.050 × 10−2 1.055 × 10−2 9.293 × 10−8 5.678 × 10−5

Errors and order, Example 6.6 (b)


64 5.207 × 10−2 5.872 × 10−2 9.875 × 10−4 8.904 × 10−3

128 4.105 × 10−2 3.725 × 10−2 3.385 × 10−4 7.262 × 10−3

256 3.173 × 10−2 2.240 × 10−2 9.798 × 10−5 8.065 × 10−3

Errors and order, Example 6.6 (c)


64 1.310 × 10−2 7.683 × 10−3 4.625 × 10−3 4.625 × 10−3

128 9.302 × 10−3 8.202 × 10−3 1.859 × 10−3 1.872 × 10−3

256 6.445 × 10−3 7.156 × 10−3 7.426 × 10−4 7.570 × 10−4

Errors and order, Example 6.6 (d)

N Narrow Elliptic (nθ = 3) Wide Elliptic (nθ = 7) Standard Filter32 3.964 × 10−2 9.813 × 10−2 - 1.926 × 10−2

64 3.788 × 10−2 4.679 × 10−2 - 8.309 × 10−3

128 3.688 × 10−2 2.367 × 10−2 - 2.482 × 10−3

256 3.003 × 10−2 1.798 × 10−2 - 9.697 × 10−4

Table 6.2: Accuracy in the l∞ norm and order of convergence of the schemes for Example 6.6 with regularizedschemes.

6.6.3 Accuracy for the time dependent problem

Recall here that for general boundary conditions, the time dependent PDE (AC) requires

Neumann boundary conditions in order for uniqueness of viscosity solutions to hold. We


have already established (in the previous section) the stability of the numerical method. In

the following examples, we test the accuracy of solutions, comparing two different wide

stencil discretizations, along with regularized filtered discretization and the (generally

unstable) standard finite difference method. Consequently we test as well the accuracy

using Dirichlet boundary conditions coming from the exact solution of Lemma 6.2.6.

Example 6.7 (Neumann BC). We consider the exact solution

u(x, y, t) = min

t +3

4

(

b

ax2 +

a

by2

)2/3

− 1, 0

.

with a = 2 and b = 1 (see Lemma 6.2.6). By taking the minimum with 0, we are imposing

homogeneous Neumann boundary conditions, thus avoiding having to deal with boundary

of the computational domain. We set Ω = [−3, 3]2.

We display the numerical error in the l∞ norm at time T = 0.1 in Table 6.3.

Example 6.8 (Dirichlet BC). We consider the exact solution

u(x, y, t) = t +3

4

(

b

ax2 +

a

by2

)2/3

.

with a = 2 and b = 1 (see Lemma 6.2.6). This is the same example as in Example 6.7, but we

consider Dirichlet boundary conditions instead. Therefore, we prescribe the exact solution

at a seven point layer at the boundary for all time t. This way all schemes are initialized at

the same set of grid points and thus we can compare their accuracy.

The error in the l∞ norm at time T = 0.1 is presented in Table 6.3.

When using Neumman boundary conditions, we observed slow convergence, with

errors near 0.01, slowly decreasing as the grid size improved. The accuracy improved as

we went from the narrow to the wider elliptic scheme, and further improved as we went

to the accurate and then the filtered scheme. In the case of Dirichlet boundary conditions,

the accuracy is better overall and the error decrease is slightly faster. The difference is

explained by the cap off done in the Neumman boundary conditions that introduces an

additional error in the solution. However, this error does not propagate to the whole

domain as the level sets of the solution shrink to its interior and so when we look at the

error away from the cap off, we recover results very similar to the Dirichlet boundary

conditions.


Errors and order, Example 6.7N Narrow Elliptic (nθ = 3) Wide Elliptic (nθ = 7) Standard Filter32 4.845 × 10−2 6.691 × 10−2 4.894 × 10−2 4.888 × 10−2

64 4.432 × 10−2 4.607 × 10−2 2.977 × 10−2 2.975 × 10−2

128 3.544 × 10−2 2.823 × 10−2 2.457 × 10−2 2.438 × 10−2

256 2.971 × 10−2 2.080 × 10−2 1.747 × 10−2 1.724 × 10−2

512 2.764 × 10−2 1.652 × 10−2 1.205 × 10−2 1.182 × 10−2

Errors and order, Example 6.8N Narrow Elliptic (nθ = 3) Wide Elliptic (nθ = 7) Standard Filter32 2.182 × 10−2 1.449 × 10−2 1.985 × 10−2 1.985 × 10−2

64 1.435 × 10−2 1.160 × 10−2 1.279 × 10−2 1.279 × 10−2

128 9.580 × 10−3 7.517 × 10−3 5.566 × 10−3 5.567 × 10−3

256 6.404 × 10−3 4.854 × 10−3 2.442 × 10−3 2.409 × 10−3

512 6.090 × 10−3 4.288 × 10−3 1.036 × 10−3 1.002 × 10−3

Table 6.3: Error in the l∞ norm of the whole computational domain at time t = 0.1 for the time dependentExample 6.7 and 6.8.

6.6.4 Numerical study of the morphology and affine invariance properties

In this section, we test if our proposed schemes satisfy numerically the morphology and

affine invariance properties that (AC) satisfies (see Theorem 6.2.5).

Example 6.9. In this example, we test numerically if the schemes presented here satisfy the

morphology property of the affine curvature evolution ((ii) in Theorem 6.2.5). We consider

two examples: (a) g1(x) = ex, (b) g2(x) = x3. We take

u0(x, y) = min

(

x

2

)2

+ y2 − 1, 0

and compare Φt(gv u0) with gv Φt(u0) at t = 1 for v = 1, 2. We took [−3, 3]2 as the

computational domain with homogeneous Neumann boundary conditions. The results

are displayed in Table 6.4. The difference in the l∞ norm is one order of magnitude smaller

than the observed accuracy for the schemes in subsection 6.6.2. Based on these examples,

the morphology property seems to hold numerically.

Example 6.10. In this example we do a qualitative test of the affine invariance property,

which in practice is what one needs for applications in image analysis. In order to do so


Difference in the l∞ norm, Example 6.9 (a)N Narrow Elliptic (nθ = 3) Wide Elliptic (nθ = 7) Standard Filter32 8.943 × 10−3 1.037 × 10−2 3.419 × 10−3 3.446 × 10−3

64 5.709 × 10−3 6.111 × 10−3 1.954 × 10−3 1.970 × 10−3

128 4.061 × 10−3 3.257 × 10−3 1.061 × 10−3 1.109 × 10−3

256 3.115 × 10−3 1.871 × 10−3 5.907 × 10−4 6.109 × 10−4

512 2.604 × 10−3 1.161 × 10−3 3.207 × 10−4 3.397 × 10−4

Difference in the l∞ norm, Example 6.9 (b)N Narrow Elliptic (nθ = 3) Wide Elliptic (nθ = 7) Standard Filter32 3.450 × 10−2 7.023 × 10−2 6.947 × 10−3 6.947 × 10−3

64 1.730 × 10−2 2.842 × 10−2 1.881 × 10−3 1.877 × 10−3

128 1.032 × 10−2 9.355 × 10−3 6.254 × 10−4 6.333 × 10−4

256 6.894 × 10−3 4.307 × 10−3 2.283 × 10−4 2.307 × 10−4

512 5.372 × 10−3 2.316 × 10−3 8.343 × 10−5 8.506 × 10−5

Table 6.4: Difference in the l∞ norm between Φt(gv u0) and gv Φt(u0) for v = 1, 2 for Example 6.9.

we plot the level sets of the affine invariant motion by curvature (AC) with

u(x, y) =(

x

2

)2

+ y2 − 1

and u φ as the initial solutions. For the affine transformations φ(x) = Ax, we consider

(a) (rotation by π/4) A =

√2

2

√2

2

−√

22

√2

2

, (b) A =

12

1

1 12

.

We take [−5, 5]2 as the computational domain on a 256 × 256 grid.

In Figure 6.7 we plot the zero level set of Φt(u φ) and(

Φt(det φ)2/3(u))

φ at t = 1. The

filtered scheme provided the best results, being indistinguishable to the naked eye. For

long time, the elliptic scheme did not provide as good results, a consequence of its lower

accuracy (see subsection 6.6.2). As for the standard finite difference scheme only in (a) the

difference is indistinguishable to the naked eye like the filtered scheme. For (b), where A

is not a special affine transformation, the difference is significant, but can be removed by

taking time steps of half the size (For shorter times all the curves were very close).

We point out that when the affine transformation is a rotation by a multiple of π/2 or

a reflection over a line L that makes an angle multiple of π/4 with the x-axis, we obtain

essentially machine precision (The difference in the l∞ norm was of the order 10−10.). This

is expected since our stencil is invariant under these transformations.

134 6.7. CONCLUSIONS

-0.5 0 0.5

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-0.5 0 0.5

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-0.5 0 0.5

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Figure 6.7: Plot of the zero level sets in Example 6.10 of Φt(u φ) and(

Φt(det φ)2/3(u))

φ for regular

elliptic scheme (left), standard scheme (center) and regular filtered scheme (right) at time t = 1 with φ givenby (a) (top) and (b) (bottom).

6.7 CONC LUS IONS

We presented a convergent finite difference discretization of the PDE for motion of level

sets by affine curvature in two dimensions. Computational examples demonstrated that the

standard finite difference method is unstable, which motivates the need for a convergent

method.

The foundation of the scheme used an existing wide stencil discretization of the mean

curvature operator, combined with an elliptic discretization of the positive and negative

eikonal operators, ±|∇u|. However, explicit time discretizations require Lipschitz continu-

ous operators, which the affine curvature operator fails to be. Thus, we approximated it by

a Lipschitz continuous regularization. In theory, the explicit Euler discretization is stable

using a time step dt ≤ O(h22/9) , with the constant determined by the width of the stencil.

In practice, we achieved numerically equivalent results using dt = h2/2 and without the

regularization, although there is no proof of stability with the less restrictive time step.

A careful choice of the regularization parameters allowed for the regularized elliptic

scheme to maintain the same order of accuracy as the unregularized scheme, while being

provably convergent. The lower accuracy of both schemes, which results from the singular-

ity of the operator, is overcomed by the use of the filtered schemes, which essentially attain

the accuracy of the standard finite difference schemes, while being provably convergent.


Simulations demonstrated the geometric properties of the PDE were nearly preserved

by the numerical solutions, including affine invariance, morphological properties, and the

accurate representation of the shrinking ellipses. Simulations validated the convergence of

the elliptic scheme, and the improved accuracy of the filtered scheme.

CHAPTER 7

CONCLUSIONS

In this thesis, we tackled three different elliptic partial differential equations (PDEs).

Filtered schemes for Hamilton-Jacobi equations were proposed, which allow us to construct

provably convergent, high order accurate finite difference schemes. For the 2-Hessian

equation in the three-dimensional case, we gave two different discretizations: a naive

one obtained by simply using standard finite differences and a monotone discretization.

The monotone discretization is provably convergent but less accurate due to the use of a

wide stencil that introduces a directional resolution error. As for the PDE that governs the

planar motion of level sets by affine curvature, we presented a convergent finite difference

discretization. A standard finite difference method was also considered, but computational

examples demonstrated its lack of stability.

Further extensions of the work can be explored. In the case of Hamilton-Jacobi equa-

tions, two fast solvers are available for monotone schemes: fast sweeping and fast marching.

In this thesis, fast sweeping solvers for the filtered schemes were proposed. A natural

extension is to develop a fast marching algorithm for filtered schemes for Hamilton-Jacobi

equations. As for the 2-Hessian equation, the schemes proposed in this thesis can be used

to build schemes for the prescribed scalar curvature of a three dimensional graph. The

reason for this is that the 2-Hessian equation is related to the scalar curvature, they are

equal up to a constant when the gradient of the function vanishes. Moreover, an extension

of the scheme for 2-Hessian equation in arbitrary dimensions can also be explored.

Finally, it is worth emphasizing that the filtered schemes are simple to implement, and

allow for an unrestricted choice of accurate schemes. It should also be clear that they

can be used for different PDEs and frameworks (e.g. discontinuous Garlekin), while still

retaining the advantages of accuracy, stability, and convergence to the viscosity solution

of the monotone schemes. However, there are still some challenges to be addressed. The

choice of the filter parameter has to be made carefully: currently, there is only a good

heuristic available. Moreover, the choice of the filter function is also worth exploring. In

the case of the Monge-Ampère equation and Hamilton-Jacobi equations, the choice was

motivated by the type of solver used, while for the affine curvature evolution a different

formulation was proposed, with the same underlying principle, which exhibits better

accuracy numerically.

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