Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Topic Course on Numerical Methods inComputational Fluid Dynamics
Lecture 2: Hyperbolic Conservation Laws
Jingmei Qiu
Department of Mathematical ScienceUniversity of Delaware
1 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Outline
1 Introduction: definition and examples
2 Development of shocks, weak solutions and the entropysolution.
3 Mathematical properties.
2 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Hyperbolic Conservation Laws
1. Definition and Examples
3 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Hyperbolic conservation laws
Integral form of conservation laws:
d
dt
∫V
udx +
∫∂V
f(u) · nds = 0,
on any control volume V .
• The integral value of u changes in time only due to thenet effect of the flux across the control volume boundary.
• If periodic or compact boundary condition:
d
dt
∫V
udx = 0.
4 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Hyperbolic conservation laws
Differential form of conservation laws:
ut +∇x · f(u) = 0, on ΩI .C . : u(x, t = 0) = u0(x), on Ω.
(1)
• u(x, t): conserved quantities
Rd × R+ → Rm,
• d : the dimension of the problem;• m: the number of components in ~u.
• f(u): flux functions
Rm → Rm
5 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Hyperbolic: 1D system
The 1-D system (1) is hyperbolic if the m×m Jacobian matrix
f ′(u) =∂f
∂u=
∂f1∂u1
· · · ∂f1∂um
· · ·∂fm∂u1
· · · ∂fm∂um
of the flux function has the following property: For each valueof u,
• the eigenvalues of f ′(u) are real,
• the matrix is diagonalizable, i.e. there is a complete set ofm linearly independent eigenvectors.
What if not real? (well-posedness) What if not diagonizable?(weakly hyperbolic)
6 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Example: d = 1, m = 1
Scalar 1D traffic flow model
∂tρ+ ∂x f (ρ) = 0,• ρ = ρ(t, x): car density.
• v(ρ): average velocity ofthe traffic,e.g.v(ρ) = vmax(ρmax − ρ).
• f (ρ) = ρv(ρ)
• Jacobina f ′(ρ) ∈ R.
7 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Example: d = 1, m = 21D shallow water system for fluid in a channel:conservation of
mass : ht + (hu)x = 0,momentum : (hu)t + (hu2 + 1
2gh2)x = 0.
(2)
• h: water height.
• u: horizontal velocity.
• 12gh
2: hydrostaticpressure from verticalintegration.
• How the equations arederived?
• What is f ′(u)? Check thesystem is hyperbolic.
8 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
1D Euler system 1
1I do like CFD. Equation of state for ideal gas law: specific total energy ρE = p
γ−1+ 1
2ρu2; specific
total enthalpy ρH = ρE + p; sound speed c =√∂p∂ρ
.
9 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Hyperbolic: high-D system
The 2-D system in the form of
ut + f(u)x + g(u)y = 0 (3)
with u: R2 × R+ → Rm, f, g : Rm → Rm is hyperbolic if anylinear combination of m×m matrix αf ′(u) + βg′(u), ∀α, β ∈ Rof the flux Jacobians has the following property: For each valueof u,
• the eigenvalues of f ′(u) are real,
• the matrix is diagonalizable, i.e. there is a complete set ofm linearly independent eigenvectors.
10 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Example: d = 2/3, m = 5
2/3D Euler system for air flow, formulated base onconservation of mass (ρ), momentum (ρu) and energy (E ).
ρt +∇ · (ρu) = 0,
(ρu)t +∇ · (ρu⊗ u) +∇p = 0,
Et +∇ · ((E + pI )u) = 0.
u = (ρ, ρu,E )T
f = (ρu, ρu⊗ u + pI,Eu + pu)
11 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
2D Euler system 2
2I do like CFD12 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
13 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
14 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
15 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Solutions of Hyperbolic Conservation Laws
2. Development of shocks, weak solutions and entropy solution.
16 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
1D scalar hyperbolic conservation law
ρt + f (ρ)x = 0.
• Characteristic line in the x − t plane,
dx
dt= f ′(ρ).
Along characteristic lines:
d
dtρ(x , t) = ρt + ρx
dx
dt= ρt + f ′(ρ)ρx = 0
Hence ρ(x , t) remains constant along characteristics.
17 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Illustration in the x − t plane
x
t
dxdt = f ′(ρ)
(0, 0)
18 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Linear advection equationf (ρ) = ρ.
• characteristics: dxdt = 1
• linear advection of initial data with speed 1.
19 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Nonlinear Burgers’ equationf (ρ) = ρ2/2.
• characteristics: dxdt = ρ(x , t).
• depending on the sign of initial data, characteristics go todifferent directions with different speeds.
• when characteristics run into each other: development ofdiscontinuities even from smooth initial data.
20 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Shock developmentExample: Burgers’ equation with sin(x) initial condition, whenthe shock will be developed?
21 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Homework
Use the method of characteristics.
22 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Weak solutions
Solutions in classical sense could fail to exist: development ofdiscontinuities or shocks in future solutions, even with smoothinitial data.
Definition: Weak solution
The function u(x , t) is called a weak solution of the con-servation laws if∫ ∞
0
∫ ∞−∞
(φtu+φx f (u))dxdt+
∫ ∞−∞
φ(x , 0)u(x , 0)dx = 0.
(4)holds for all test function φ ∈ C 1
0 (R×R+), where C 10 is
the space of function that are continuously differentiablewith compact support.
23 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Mathematically the definition of weak solution in the integralform above is equivalent to the solution for the integral form ofthe equation over any choices of spatial and time interval.
• Finite volume scheme: formulated base on the integralform.
• Discontinuous Galerkin method: formulated base onthe weak form (test functions etc.).
24 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Weak solution: R-H jumpcondition
Proposition: Rankine-Hugoniot condition
A function u(x , t) is piecewise smooth and satisfies thePDE strong whenever u ∈ C 1. If the function satisfies theRankine-Hugoniot jump condition along the discontinuitycurve,
s.
= x ′(t) =[|f (u)|]
[|u|], (5)
then u(x , t) is a weak solution of the nonlinear hyperbolicequation. s
.= x ′(t) is the shock speed.
25 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Illustration in the x − t plane
x
tShock
a[
b]
u− u+
x− x+
(0, 0)
26 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Proof base on the integral form
27 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Riemann Problem
• A Riemann problem consists of a conservation lawtogether with piecewise constant initial data having asingle discontinuity.
• The exact solution, involving shocks and rarefactionwaves, to nonlinear Euler equation with Riemann initialdata, can be analytically derived.
• In numerical analysis, Riemann problems appear in anatural way in finite volume methods for the solution ofequation of conservation laws. It is widely used incomputational fluid dynamics and in MHD simulations. Inthese fields Riemann problems are calculated using(approximate) Riemann solvers.
28 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Riemann problem: Example 1
Consider Burgers’ equation, ut + (u2
2 )x = 0 with the initialcondition
u0(x) =
ul = 1, x < 0ur = −1, x > 0
(6)
• Drawing characteristics: colliding into each other(formation of shocks).
• By the R-H jump condition, u(x , t) = u0(x) is the weaksolution of the Burgers’ equation.
29 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Illustration in the x − t plane
x
t
dxdt = 1 dx
dt = −1
(0, 0)
Shock
30 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Riemann problem: Example 2 andnon-uniqueness of weak solution
Consider Burgers’ equation with the initial condition
u0(x) =
ul = −1, x < 0ur = 1, x > 0
(7)
• Drawing characteristics: spreading out.
• A weak solution is the rarefaction wave
u(x , t) =
−1, x < −txt , −t < x < t1, x > t
(8)
• How to derive rarefaction wave solution in the middle?
31 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Illustration in the x − t plane
x
t
Rarefaction
dxdt = −1 dx
dt = 1
(0, 0)
32 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
On the other hand,
• By R-H jump condition, u(x , t) = u0(x) is also a weaksolution of the Burgers’ equation.
• In fact, there are infinitely many weak solutions for thisproblem.
Uniqueness of the weak solution is lost.
33 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Homework
Use the method of characteristics and R-H condition.
34 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Entropy solution
The uniquely exist physically relevant solution among weaksolutions.
• How to define entropies solutions? (vanishing viscositymethod, entropy inequalities)
• What are criteria for selecting entropy solutions amongweak solutions? (entropy conditions)
• What are appropriate spaces (norms) for entropysolutions? (BV norm, L1 and L∞)
35 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Entropy solution
In physical models, the balance laws only come with somephysical viscosity. Conservation laws with viscous terms providemore physically relevant models. For example, in the trafficflow modeling, the ”viscosity” takes the form of slow responseof drivers and automobiles; in the fluid dynamics, the viscositycorresponds to the informal notion of ”thickness”. Forexample, honey has a higher viscosity than water.
36 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Entropy solution: vanishingviscosity method
Definition: Vanishing viscosity principle
Consider the viscous equation
uεt + f (uε)x = εuεxx (9)
An entropy solution of the nonlinear hyperbolic equationis the limit (a.e.) of uε of equation (9) when ε→ 0.
• uε exists for the viscous equation (9).
• The limit of uε exists, as ε goes to zero.
37 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Entropy solution: entropyinequality
Definition: Entropy solution
A weak solution of the nonlinear scalar conservation lawsut + f (u)x = 0 is an entropy solution if for all convexentropy function U(u) with U ′′(u) ≥ 0 and the associatedentropy flux function F (u) with F ′(u) = U ′(u)f ′(u), wehave
U(u)t + F (u)x ≤ 0, (10)
in the distribution sense.
• That is for all φ ≥ 0, φ ∈ C 10 (R× R+), we have
−∫∞
0
∫∞−∞(φtU(u) + φxF (u))dxdt
−∫∞−∞ φ(x , 0)U(u(x , 0))dx ≤ 0.
38 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
39 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
R-H inequality across shock
In particular, across discontinuities in the x − t plane, we haveRankin-Hugoniot like entropy inequality,
−s[|U|] + [|F (U)|] ≤ 0,
for any entropy-entropy flux pairs.
Recall that R-H jump condition across shock is
−s[|u|] + [|f (u)|] = 0.
40 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Example.
Consider Burgers’ equation with Riemann initial data.— Which one is the entropy solution? Rarefaction wave orshock?
41 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Oleinik entropy condition
A discontinuity propagating with speed s = [|f (u)|][|u|] given by the
Rankine-Hugoniot jump condition satisfies the Oleinik entropycondition if for all u between ul and ur ,
f (u)− f (ul)
u − ul≥ s ≥ f (u)− f (ur )
u − ur(11)
where ul and ur are left and right state along the discontinuityrespectively.
Example. Rarefaction wave vs. shock for Riemann problem:which one is the entropy solution?
42 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Lax entropy condition
A discontinuity propagating with speed s given by theRankine-Hugoniot jump condition satisfies the Lax entropycondition if
f ′(ul) > s > f ′(ur ), (12)
where ul and ur are left and right state along the discontinuityrespectively.
Example. Rarefaction wave vs. shock for Riemann problem:which one is the entropy solution?
43 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Entropy Solution ofHyperbolic Conservation Laws
3. Mathematical properties:well-posedness in BV, L1 and L∞ norms.
44 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Bounded Variation (BV) norm
• A function is of bounded variation on a given interval[a, b], if
V ba (f )
.= sup
∑i
|f (xi )− f (xi+1)| <∞,
where xii is any partition of [a, b].
• If u is continuously differentiable, then
V ba (u)
.=
∫ b
a|u′(x)|dx .
• BV norm is an important concept for numerical schemesfor nonlinear equations: TVD, TVB, ENO, WENOschemes.
• Numerical oscillations will contribute to extra functionvariations that are not physical.
45 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Well-posedness in BoundedVariation (BV) norm
Proposition. If u0(x) is a function of locally bounded variationon (−∞,∞), then for each t > 0, u(·, t) is also a function oflocally bounded variation on (−∞,∞), and
TV R−Ru(·, t) ≤ TV R+st
−R−stu0(·),
where s = maxx |f ′(u)|.
46 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
L1 contraction property
Proposition. If u(x , t) and v(x , t) are solutions of the scalarhyperbolic equation with initial data u0(x) and v0(x)respectively, then
‖u(·, t)− v(·, t)‖L1 ≤ ‖u0(·)− v0(·)‖L1 .
Specifically, consider v ≡ 0, then ‖u(·, t)‖L1 ≤ ‖u0(·)‖L1 .
47 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
48 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
49 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
L∞ maximum principle
Proposition. If u(x , t) is a solution of the scalar hyperbolicequation with initial condition u0(x), then
maxx
u(x , t) ≤ maxx
u0(x), minx
u(x , t) ≥ minx
u0(x),
50 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
1D system and high D problems5
• 1D hyperbolic system:— Riemann problem: Hugoniot locus and integral curves.— well-posedness results available only for special class ofinitial data, i.e. initial data with small enough totalvariation. (counter example exists when total variation ofthe solution grows geometrically.3)
• high-D scalar and systems. 4
3https://www.math.psu.edu/bressan/PSPDF/claw-questions.pdf4Measure-Theoretic Analysis and Nonlinear Conservation Laws, Chen,
Torres and Ziemer, 20075https://www.math.psu.edu/bressan/PSPDF/claw-questions.pdf
51 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
Summary
Summary
• Characteristics method: before shocks
• Weak solutions and Rankin-Hugoniot jump condition.
• Riemann problems.
• Entropy solution and entropy inequality
• Well-posedness in bounded variation norm, L1 norm andL∞ norm.
52 / 53
Comp. Math.&
Applications
Jingmei Qiu
Outline
HCL
SummaryAcknowledgement:
• Lecture notes by Chi-Wang Shu, when I was a graduatestudent at Brown University.
• Special thanks to Ms. Mingchang Ding (Ph.D. student atUniversity of Delaware) for her help in preparing theseslides.
53 / 53
