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Numerical Methods for Differential Equations

Chapter 5: Partial differential equations – elliptic and pa rabolic

Gustaf Soderlind and Carmen Arevalo

Numerical Analysis, Lund University

Textbooks: A First Course in the Numerical Analysis of Differential Equations, by Arieh Iserles

and Introduction to Mathematical Modelling with Differential Equations, by Lennart Edsberg

c© Gustaf Soderlind, Numerical Analysis, Mathematical Sciences, Lund University, 2008-09

Numerical Methods for Differential Equations – p. 1/50

1. Brief overview of PDE problems

Classification: Three basic types, four prototype equations

Elliptic

∆u = 0 + BC

Parabolic

ut = ∆u + BC & IC

Hyperbolic

utt = ∆u + BC & IC

ut + a(u)ux = 0 + BC & IC

Numerical Methods for Differential Equations – p. 2/50

Classification of PDEs

Linear PDE with two independent variables

Auxx + 2Buxy + Cuyy + L(ux, uy, u, x, y) = 0

with L linear in ux, uy, u. Study

δ := det

A B

B C

= AC − B2

Elliptic δ > 0 Parabolic δ = 0 Hyperbolic δ < 0

Numerical Methods for Differential Equations – p. 3/50

Standard PDEs. Prototypical problems

δ > 0 Elliptic PDE Laplace equation

uxx + uyy = 0 A = C = 1; B = 0

δ = 0 Parabolic PDE Diffusion equation

ut = uxx A = 1; B = C = 0

δ < 0 Hyperbolic PDEs Wave equation

utt = uxx A = 1; B = 0; C = −1

Numerical Methods for Differential Equations – p. 4/50

PDE method types

FDM Finite difference methods

FEM Finite element methods

FVM Finite volume methods

BEM Boundary element methods

We will mostly study FDM to cover basic theory

Industrial relevance: FEM

Numerical Methods for Differential Equations – p. 5/50

PDE methods for elliptic problems

Simple geometry FDM or Fourier methods

Complex geometry FEM

Special problems FVM or BEM

Large sparse systemsCombine with iterative solvers such as multigrid methods

Numerical Methods for Differential Equations – p. 6/50

PDE methods for parabolic problems

Simple geometry FDM or Fourier methods

Complex geometry FEM

StiffnessAlways use A–stable time-stepping methods

Need Newton-type solvers for large sparse systems

Numerical Methods for Differential Equations – p. 7/50

PDE methods for hyperbolic problems

FDM, FVM. Sometimes FEM

ShocksSolutions may be discontinuous – example: “sonic boom”

TurbulenceMultiscale phenomena

Hyperbolic problems have several complications and manyhighly specialized techniques are often needed

Numerical Methods for Differential Equations – p. 8/50

2. Elliptic problems. FDM

Laplacian ∆ =∂2

∂x2+

∂2

∂y2+

∂2

∂z2

Laplace equation ∆u = 0

with boundary conditions u = u0(x, y, z) x, y, z ∈ ∂Ω

Poisson equation −∆u = f

with boundary conditions u = u0(x, y, z) x, y, z ∈ ∂Ω

Other boundary conditions also of interest (∂Ω = bdry of Ω)

Numerical Methods for Differential Equations – p. 9/50

Elliptic problems. Some applications

Equilibrium problems

Structural analysis (strength of materials)

Heat distribution

Potential problems

Potential flow (inviscid, subsonic flow)

Electromagnetics (fields, radiation)

Eigenvalue problems

Acoustics

Microphysics

Numerical Methods for Differential Equations – p. 10/50

Poisson equation – an elliptic model problem

∂2u

∂x2+

∂2u

∂y2= f(x, y)

Computational domain Ω = [0, 1] × [0, 1] (unit square)Dirichlet conditions u(x, y) = 0 on boundary

Uniform grid xi, yjN,Mi,j=1 with equidistant mesh widths

∆x = 1/(N + 1) and ∆y = 1/(M + 1)

Discretization Finite differences with ui,j ≈ u(xi, yj)

ui−1,j − 2ui,j + ui+1,j

∆x2+

ui,j−1 − 2ui,j + ui,j+1

∆y2= f(xi, yj)

Numerical Methods for Differential Equations – p. 11/50

Equidistant mesh ∆x = ∆y

ui−1,j + ui,j−1 − 4ui,j + ui,j+1 + ui+1,j

∆x2= f(xi, yj)

Participating approximations and mesh points

xi−1 xi xi+1

yj

yj+1

yj−1

Numerical Methods for Differential Equations – p. 12/50

Computational “stencil” for ∆x = ∆y

ui−1,j + ui,j−1 − 4ui,j + ui,j+1 + ui+1,j

∆x2= f(xi, yj)

“Five-point operator”1

1 −4 1

1

Numerical Methods for Differential Equations – p. 13/50

The FDM linear system of equations

Lexicographic ordering of unknowns ⇒ partitioned system

1

∆x2

T I 0 . . .

I T I

I T I

. . . I

. . . 0 I T

u·,1

u·,2

u·,3

...

u·,N

=

f(x·, y1)

f(x·, y2)

f(x·, y3)...

f(x·, yN )

with Toeplitz matrix T = tridiag(1 −4 1)

The system is N 2 × N 2, hence large and very sparse

Numerical Methods for Differential Equations – p. 14/50

3. Elliptic problems. FEM

Finite Element Method

PDE Lu = 0

Ansatz u =∑

ciϕi ⇒ Lu =∑

ciLϕi

Requirement 〈ϕi, Lu〉 = 0 gives coefficients ci

FEM is a least squares approximation, fitting a linearcombination of basis functions ϕi to the differentialequation using orthogonality

Simplest case Piecewise linear basis functions

Numerical Methods for Differential Equations – p. 15/50

Strong and weak forms

Strong form

−∆u = f ; u = 0 on ∂Ω

Take v with v = 0 on ∂Ω∫

Ω

−∆u · v =

∫

Ω

f · v

Integrate by parts to get weak form∫

Ω

∇u · ∇v =

∫

Ω

f · v

Numerical Methods for Differential Equations – p. 16/50

Strong and weak forms. 1D case

Recall integration by parts in 1D

∫ 1

0

−u′′v = [−u′v]10 +

∫ 1

0

u′v′

or in terms of an inner product

−〈u′′, v〉 = 〈u′, v′〉

Generalization to 2D, 3D uses vector calculus

Numerical Methods for Differential Equations – p. 17/50

Weak form of −∆u = f

Define inner product

〈v, u〉 =

∫

Ω

vu dΩ

and energy norm (note scalar product!)

a(v, u) =

∫

Ω

∇v · ∇u dΩ

to get the weak form of −∆u = f as

a(v, u) = 〈v, f〉

Numerical Methods for Differential Equations – p. 18/50

Galerkin method (Finite Element Method)

1. Basis functions ϕi2. Approximate u =

∑

cjϕj

3. Determine cj from∑

cj

∫

∇ϕi · ∇ϕj =∫

fϕi

The cj are determined by the linear system

Kc = F

The matrix K is called stiffness matrix

Stiffness matrix elements kij =∫

∇ϕi · ∇ϕj = a(ϕi, ϕj)

Right-hand side Fi =∫

ϕif = 〈ϕi, f〉

Numerical Methods for Differential Equations – p. 19/50

The FEM mesh. Domain triangulation

Piecewise linear basis ϕj require domain triangulation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Numerical Methods for Differential Equations – p. 20/50

4. Parabolic problems

The prototypical equation is the

Diffusion equation ut = ∆u

Nonlinear diffusion

ut = div (k(u)gradu)

Boundary and initial conditions are needed

Numerical Methods for Differential Equations – p. 21/50

Parabolic problems. Some applications

Diffusive processes

Heat conduction ut = d · uxx

Chemical reactions

Reaction–diffusion ut = d · uxx + f(u)

Convection–diffusion ut = ux + 1Pe

uxx

Irreversibility ut = −∆u is not well-posed!

Numerical Methods for Differential Equations – p. 22/50

Diffusion – a parabolic model problem

Equation ut = uxx

Initial values u(x, 0) = g(x)

Boundary values u(0, t) = u(1, t) = 0

Separation of variables u(x, t) := X(x)T (t) ⇒

ut = XT , uxx = X ′′T ⇒ T

T=

X ′′

X=: λ

T = Ceλt X = A sin√−λx + B cos

√−λx

Numerical Methods for Differential Equations – p. 23/50

Parabolic model problem. . .

Boundary values

X(0) = X(1) = 0 ⇒ λk = −(kπ)2, therefore

Xk(x) =√

2 sin kπx Tk(t) = e−(kπ)2t

Initial values

Fourier expansion g(x) =∑

∞

1 ck

√2 sin kπx ⇒

u(x, t) =√

2

∞∑

k=1

cke−(kπ)2tsin kπx

Numerical Methods for Differential Equations – p. 24/50

5. Method of lines (MOL) discretization

In ut = uxx, discretize ∂2/∂x2 by

uxx ≈ ui−1 − 2ui + ui+1

∆x2

System of ODEs (semidiscretization) u = T∆xu

u =1

∆x2

−2 1

1 −2 1. . .

1 −2

u

Numerical Methods for Differential Equations – p. 25/50

Full FDM discretization

Note that ui(t) ≈ u(x, t) along the line x = xi in (x, t) plane

Use time-stepping to solve the IVP; Explicit Euler withui,j ≈ u(xi, tj) implies

ui,j+1 − ui,j

∆t=

ui−1,j − 2ui,j + ui+1,j

∆x2

With the Courant number µ = ∆t/∆x2 we obtain recursion

ui,j+1 = ui,j + µ · (ui−1,j − 2ui,j + ui+1,j)

Numerical Methods for Differential Equations – p. 26/50

Method of lines. The grid

Rectangular grid (i · ∆x, j · ∆t), i = 0 : N + 1, j ≥ 0 with∆x = 1/(N + 1)

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

8

t

x

ui,j ≈ u(i · ∆x, j · ∆t)

Numerical Methods for Differential Equations – p. 27/50

Method of lines. Computational stencil

Explicit Euler time stepping. Participating grid points

xi−1 xi xi+1

tj

tj+1

Courant number µ = ∆t/∆x2

1

−µ 2µ−1 −µ

Numerical Methods for Differential Equations – p. 28/50

Method of lines. Stability and the CFL condition

Explicit Euler with ∆x = 1/(N + 1) implies recursion

u·,j+1 = u·,j + ∆t · T∆xu·,j

Recall λk[T∆x] = −4(N + 1)2 sin2 kπ

2(N + 1)for k = 1 : N

Stability requires ∆t · λk ∈ S for all eigenvalues

−2.5 −2 −1.5 −1 −0.5 0 0.5−1.5

−1

−0.5

0

0.5

1

1.5

Re

Im

∆t · λk ∈[

− 4∆t

∆x2,−π2∆t

]

S

Numerical Methods for Differential Equations – p. 29/50

The CFL condition

For stability we need 4∆t/∆x2 ≤ 2

CFL condition (Courant, Friedrichs, Lewy 1928)

∆t

∆x2≤ 1

2

The CFL condition is a severe restriction on time step ∆t

Stiffness The CFL condition can be avoided by usingA-stable methods, e.g. Trapezoidal Rule or Implicit Euler

Numerical Methods for Differential Equations – p. 30/50

Experimental stability investigation

N = 30 internal pts in [0, 1], M = 187 time steps on [0, 0.1]

Stable solution at CFL = .514

00.02

0.040.06

0.080.1

0

0.2

0.4

0.6

0.8

10

0.2

0.4

0.6

0.8

1

Numerical Methods for Differential Equations – p. 31/50

Violating the CFL condition. Instability

N = 30 internal pts in [0, 1], M = 184 time steps on [0, 0.1]

Unstable solution at CFL = .522

00.02

0.040.06

0.080.1

0

0.2

0.4

0.6

0.8

1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Numerical Methods for Differential Equations – p. 32/50

Crank–Nicolson method (1947)

Crank–Nicolson method ⇔ Trapezoidal Rule for PDEs

The trapezoidal rule is

implicit ⇒ more work/step

A–stable ⇒ no restriction on ∆t

Theorem Crank–Nicolson is unconditionally stable

There is no CFL condition on the time-step ∆t

Numerical Methods for Differential Equations – p. 33/50

Crank–Nicolson method. . .

Courant number µ = ∆t/∆x2 ⇒ recursion

(I − µ

2T )u·,j+1 = (I +

µ

2T )u·,j

with Toeplitz matrix T = tridiag(1 − 2 1)

Tridiagonal structure ⇒ low complexity

Refactorize only if Courant number µ = ∆t/∆x2 changes

Numerical Methods for Differential Equations – p. 34/50

6. Error analysis. Convergence

MOL with explicit Euler for ut = uxx

Global error ei,j = ui,j − u(xi, tj)

Local error Insert exact solution to get

u(xi, tj+1) − u(xi, tj)

∆t=

=u(xi−1, tj) − 2u(xi, tj) + u(xi+1, tj)

∆x2− li,j

Expand local error li,j in Taylor series

Numerical Methods for Differential Equations – p. 35/50

Local error

Taylor expansion

−li,j = ut − uxx +∆t

2utt +

∆x2

12uxxxx + O(∆t2,∆x4)

Therefore

−li,j =∆t

2utt +

∆x2

12uxxxx + O(∆t2,∆x4)

Numerical Methods for Differential Equations – p. 36/50

The Lax Principle

Conclusion

Consistency li,j → 0 as ∆t,∆x → 0

Stability CFL condition ∆t/∆x2 ≤ 1/2

Convergence ei,j → 0 as ∆t,∆x → 0

Theorem (Lax Principle)

Consistency + Stability ⇒ Convergence

Note Choice of norm is very important

Numerical Methods for Differential Equations – p. 37/50

The order of the method

With local error

−li,j =∆t

2utt +

∆x2

12uxxxx = O(∆t,∆x2)

and stability in terms of CFL condition µ = ∆t/∆x2 ≤ 1/2

we have global error ei,j = O(∆t,∆x2)

For fixed µ we have ∆t ∼ ∆x2 and it follows that

Global error ei,j = O(∆t,∆x2) = O(∆x2) ⇒

Theorem The order of convergence is p = 2

Numerical Methods for Differential Equations – p. 38/50

Order of PDE discretizations

Discretization process from PDE to SD to FD

ut = uxx → v∆x = P∆xv∆x + h∆x(t) → uj+1µ = Aµu

jµ + kj

µ

Suppose

order of SD scheme for spatial variables is p1

order of ODE time discretization is p2

Theorem If ∆t = µ∆x2 the FD order of convergence isp = minp1, 2p2

Numerical Methods for Differential Equations – p. 39/50

Example. The Crank–Nicolson method

From Trapezoidal rule yn+1 = yn + h2[f(yn) + f(yn+1)]

uj+1i = uj

i +∆t

2∆x2[(uj

i−1 − 2uji + uj

i+1) + (uj+1i−1 − 2uj+1

i + uj+1i+1 )]

−µ

2uj+1

i−1 + (1 + µ)uj+1i − µ

2uj+1

i+1 =µ

2uj

i−1 + (1 − µ)uji +

µ

2uj

i+1

Same order p = min2, 4 = 2 as with Explicit Euler

xi−1 xi xi+1

tj

tj+1

Numerical Methods for Differential Equations – p. 40/50

Crank–Nicolson. Stability

Aµ = (I − µ

2T )−1(I +

µ

2T ) with usual Toeplitz matrix T

Theorem The eigenvalues are λ[Aµ] =1 + µ

2λ[T ]

1 − µ

2λ[T ]

Note λ[T ] ∈ (−4, 0) ⇒ −1 < λ[Aµ] < 1 This implies thatthere is no CFL stability condition on the Courant ratio µ!The method is stable for all ∆t > 0

Theorem Crank–Nicolson is unconditionally stable

Numerical Methods for Differential Equations – p. 41/50

Experimental stability investigation

N = 30 internal pts in [0, 1], M = 30 time steps on [0, 0.1]

Stable solution at CFL = 3.2

00.02

0.040.06

0.080.1

0

0.2

0.4

0.6

0.8

10

0.2

0.4

0.6

0.8

1

Numerical Methods for Differential Equations – p. 42/50

7. Parabolic problems. FEM

Consider diffusion problem in strong form ut − uxx = 0 withDirichlet boundary conditions

Multiply by test function v and integrate by parts

∫ 1

0

vut dx +

∫ 1

0

v′u′ dx = 0

In terms of inner product and energy norm –

Weak form 〈v, ut〉+a(v, u) = 0 for all v with v(0) = v(1) = 0

Numerical Methods for Differential Equations – p. 43/50

Galerkin method (Finite Element Method)

1. Basis functions ϕi2. Approximate u(t, x) =

∑

cj(t)ϕj(x)

3. Determine cj from 〈ϕi, ut〉 + a(ϕi, u) = 0

Note 〈ϕi, ut〉 =∑

cj〈ϕi, ϕj〉 and a(ϕi, u) =∑

cj〈ϕ′

i, ϕ′

j〉

We get an initial value problem

B∆xc + K∆xc = 0

for the determination of the coefficients cj(t) with c(0)

determined by the initial condition

Numerical Methods for Differential Equations – p. 44/50

Galerkin method

Simplest casePiecewise linear basis functions on equidistant grid

Stiffness matrix elements kij = 〈ϕ′

i, ϕ′

j〉

K∆x =1

∆xtridiag(−1 2 −1)

and mass matrix elements bij = 〈ϕi, ϕj〉

B∆x =∆x

6tridiag(1 4 1)

Numerical Methods for Differential Equations – p. 45/50

Simplest Galerkin FEM. . .

Note that in the initial value problem

B∆xc + K∆xc = 0

the matrix B∆x is tridiagonal ⇒ no advantage fromexplicit time stepping methods

Explicit Euler

B∆x(cn+1 − cn) = −∆t · K∆xcn

requires the solution of a tridiagonal system on every step

Numerical Methods for Differential Equations – p. 46/50

Simplest Galerkin FEM. . .

As the system is stiff, consider implicit A-stable method

B∆x(cn+1 − cn) = − ∆t

2· K∆x(cn + cn+1)

and solve tridiagonal system

(B∆x +∆t

2K∆x)cn+1 = (B∆x −

∆t

2K∆x)cn

on every step

Trapezoidal rule has same cost, but better stability

Numerical Methods for Differential Equations – p. 47/50

8. Well-posedness

Linear partial differential equationut = Lu + f, 0 ≤ x ≤ 1, t ≥ 0, u(x, 0) = h(x),

u(0, t) = φ0(t), u(1, t) = φ1(t)

Suppose

wt = Lw + f, w(x, 0) = h(x)

vt = Lv + f, v(x, 0) = h(x) + g(x)

Subtract to get homogeneous Dirichlet problem

ut = Lu, u(x, 0) = g(x), φ0(t) ≡ 0, φ1(t) ≡ 0

Numerical Methods for Differential Equations – p. 48/50

Well-posedness. Time evolution

Suppose time evolution u(x, t) = E(t)g(x)

Definition The equation is well-posed if for every t∗ > 0

there is a constant 0 < C(t∗) < ∞ such that‖E(t)‖ ≤ C(t∗) for all 0 ≤ t ≤ t∗

Theorem A well-posed equation has a solution that

depends continuously on the initial value (the “data”)

is uniformly bounded in any compact interval

Numerical Methods for Differential Equations – p. 49/50

ut = uxx is well posed

Fourier series expansion g(x) =√

2∑

∞

1 ck sin kπx implies

u(x, t) =√

2∞

∑

k=1

cke−(kπ)2t sin kπx

‖E(t)g‖22 =

∫ 1

0

|u(x, t)|2 dx

= 2∞

∑

k=1

∞∑

j=1

ckcje−(k2+j2)π2t

∫ 1

0

sin kπx sin jπx dx

=∞

∑

k=1

c2ke

−2(kπ)2t ≤∞

∑

k=1

c2k = ‖g‖2

2

Hence ‖E(t)‖2 ≤ 1 for every t ≥ 0Numerical Methods for Differential Equations – p. 50/50

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