topic course on numerical methods in computational fluid

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes

NonlinearTVD andTVB schemes

Summary

Topic Course on Numerical Methods inComputational Fluid Dynamics

Lecture 4: Finite volume schemesfor 1D nonlinear scalar equations

Jingmei Qiu

Department of Mathematical SciencesUniversity of Delaware

1 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Outline

1 Nonlinear equations.

2 Conservative schemes and weak convergence.

3 Monotone schemes and entropy convergence

4 Total Variation Dimishing (TVD) schemes

5 Total Variation Bounded (TVB) schemes

2 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

1. Nonlinear Equations

3 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Scalar 1D equation

{ut + f (u)x = 0,u(x , t = 0) = u0(x).

• u: conserved variable.

• f (u): flux function.

• For simplicity: periodic boundary condition.

4 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Burgers’ equation

ut +

(u2

2

)x

= 0 (1)

• Smooth initial data, and shock development.

u0(x) = sin(x)

• Riemann initial data:

u0(x) =

{ul , x < 0ur , x ≥ 0

5 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Gain insights on solutions

Method of characteristics:

• solutions stay constant along characteristics

• Rankine-Hugoniot jump condition across shocks for weaksolutions

• Entropy conditions (inequalities) across shocks for theentropy solution

6 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Development of shocks

7 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Shocks and rarefaction waves

• Understanding Riemann solutions is a basic building blockfor finite volume (WENO) schemes for scalar and systemsof hyperbolic conservation laws.

• Depending on ul and ur , there could be a shock or ararefaction wave for Burgers’ equation (convex).

• For more general problems, Riemann solution could involvecompound waves (a combination of shocks and rarefactionwaves).1

• For systems (e.g. Navier-Stokes), there will be a family ofwaves (shock, rarefaction waves, contact discontinuity). 2

1Chap 4, LeVeque, Numerical Methods for Conservation Laws.2Chap 7-9, LeVeque, Numerical Methods for Conservation Laws.

8 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Recall well-posedness propertiesProposition

• Bounded variation

V ba (u) := sup

∑i

|u(xi )− u(xi+1)| <∞, for any partition {xi}.

V R−R u(·, t) ≤ V R+st

−R+st u0(x), s = max |f ′(u)|.

• L1 contraction property. If u(x , t) and v(x , t) are solutions ofut + f (u)x = 0 with initial data u0(x) and v0(x).

‖u(x , t)− v(x , t)‖L1 ≤ ‖u0(x)− v0(x)‖L1 .

• L∞ maximum principle. If u(x , t) is a solution of scalarconservation laws with initial data u0(x), then ∀x , t

max u0(x) ≥ max u(x , t),

min u0(x) ≤ min u(x , t).

At the discrete level, we would like to preserve these properties as much as

possible.9 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Road Map






10 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

2. Conservative Schemes

11 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

An example of upwind scheme• Consider Burgers’ equation in the quasilinear form

ut + uux = 0,

assume u0(x) ≥ 0 (hence u(x , t) ≥ 0).• An upwind scheme is

un+1j = un

j −∆t

∆xun

j (unj − un

j−1).

• Consider the following Riemann data

u0(x) =

{1, x < 00, x ≥ 0

.

Then, it can be found that

unj = u0

j (∀j , n),

which is not a weak solution of (1) (does not satisfy theR-H condition).

12 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Message:Upwind principle is NOT sufficient to guarantee weakconvergence, not to mention entropy convergence!

13 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Recall derivation of conservationlaws

ut + f (u)x = 0. (2)

∫ ba

(2) dx−−−−−→ d

dt

(∫ b

a

u(x , t)dx

)+ f (u(b, t))− f (u(a, t)) = 0.

Spatial partition: [0, 1] =N⋃

j=1

Ij with Ij = [xj− 12, xj+ 1

2].

Let [a, b] = Ij ,

d

dt

(∫Ij

u(x , t)dx

)+ f (u(xj+ 1

2, t))− f (u(xj− 1

2, t)) = 0.

14 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Finite volume schemes

1

∆x

[d

dt

(∫Ij

u(x , t)dx

)+ f (u(xj+ 1

2, t))− f (u(xj− 1

2, t))

]= 0.

Then,d

dtuj (t) +

1

∆x(fj+ 1

2− fj− 1

2) = 0

with

uj (t) ≈ 1

∆x

∫Ij

u(x , t)dx .

Thus,

un+1j = un

j −∆t

∆x(fj+ 1

2− fj− 1

2)

withfj− 1

2= f (un

j−1, unj ),

being an approximation of the Riemann solution with left and right

states unj−1, u

nj respectively.

15 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Conservative scheme

Definition (Conservative schemes)

A scheme is called a conservative scheme if it can be written in the formof

un+1j = un

j −∆t

∆x(fj+ 1

2− fj− 1

2)

where fj± 12

are numerical fluxes. They are designed to locally depends on

the solutionfj+ 1

2= f (u−

j+ 12

, u+

j+ 12

).

The flux satisfies

• f is Lipschitz continuous w.r.t. u− and u+.

• f (u, u) = f (u) (consistency).

RemarkWe say f is Lipschitz continuous if there is a constant L ≥ 0(which may depend

on u) such that |f (v , u)− f (u, u)| ≤ L|v − u|, ∀v with |v − u| is sufficiently small.

16 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Local mass conservationProposition (Mass conservation)

Mass is conserved ∑j

un+1j =

∑j

unj ,

if periodic or compact boundary condition is imposed.

∑j

un+1j =

∑j

(un

j −∆t

∆x(fj+ 1

2− fj− 1

2)

),

=∑

j

unj −

∆t

∆x

∑j

(fj+ 1

2− fj− 1

2

)=∑

j

unj .

Conservation laws are respect!17 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Lax-Wendroff Theorem

Theorem (Lax-Wendroff)

Assume the solution {unj } to a conservative scheme con-

verges (as ∆t,∆x → 0), bounded a.e. to a functionu(x , t). Then u is a weak solution to the nonlinear hy-perbolic equation.

18 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

The example of the upwind scheme for the Burgers equation,

un+1j = un

j −∆t

∆xun

j (unj − un

j−1).

is not a conservative scheme; and does not converge to weaksolution.

19 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Numerical fluxes

fj− 12

= f (u−j− 1

2

, u+j− 1

2

) = f (uj−1, uj ),

fj+ 12

= f (u−j+ 1

2

, u+j+ 1

2

) = f (uj , uj+1),

as approximate Riemann solvers.

In the following, for simplicity

f = f (u−, u+).

20 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Examples: Godunov scheme

The numerical flux of the Godunov scheme can be written as

f =

minu−≤u≤u+

f (u), if u− ≤ u+

maxu+≤u≤u−

f (u), if u− > u+ .

• Exact solution of the Riemann problem at cell interfaces.

21 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Examples: Lax-Friedrich schemeThe numerical flux is

f =1

2

[f (u−) + f (u+)− α(u+ − u−)

],

where α = maxu|f ′(u)|.

Ideas behind:

f (u) = f + + f −

:=1

2(f (u) + αu) +

1

2(f (u)− αu),

fupwind

=1

2(f (u−) + αu−) +

1

2(f (u+)− αu+)

observing that∂f +

∂u≥ 0,

∂f −

∂u≤ 0.

22 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Examples: local Lax-Friedrichscheme

The numerical flux is

f =1

2

[f (u−) + f (u+)− α(u+ − u−)

],

whereα = max

(u−,u+)|f ′(u)|.

(u−, u+) is a non-empty interval no matter which end of theinterval is greater.

23 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Examples: Roe scheme


f =

{f (u−), if a ≥ 0f (u+), if a < 0

,

where

a =

{f (u+)−f (u−)

u+−u− , if u+ 6= u−

f ′(u) if u+ = u− = u

is the speed of the solution.

24 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Examples: Engquist-Osher scheme


f = f +(u−) + f −(u+),

where

f +(u−) =

∫ u−

0max(f ′(u), 0)du + f (0),

f −(u+) =

∫ u+

0min(f ′(u), 0)du.

25 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

An example of Roe scheme

Consider the Riemann initial data

u0(x) =

{−1, x < 0

1, x ≥ 0.

Recall the conservative scheme

un+1j = un

j −∆t

∆x(fj+ 1

2− fj− 1

2)

with Roe flux,

f =1

2, everywhere.

Then,un+1

j = unj , ∀j , n.

Thus, the numerical solution stays stationary u(x , t) = u0(x),which is a weak solution, thanks to Lax-Wendroff Theorem.

26 / 79

Comp. Math.&

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Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

However, it is not an entropy solution. The entropy solutionshould be a rarefaction wave

u0(x) =

−1, x ≤ t,

xt , −t ≤ x ≤ t,1, x > t

.

Conservative scheme is sufficient to guarantee weakconvergence, but not entropy convergence!

Question: how to guarantee not only convergence to the weaksolution, but also to the unique physically relevant entropysolution?

27 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Conservative schemes: summary

Given {uj}j

1 Evaluate numerical fluxes

fj+ 12

= f (u−j+ 1

2

, u+j+ 1

2

)first order

= f (uj , uj+1), ∀j .

2 Evolve the ODE system by the explicit forward Eulermethod

un+1j = un

j −∆t

∆x

(fj+ 1

2− fj− 1

2

).

28 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Road Map






29 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

3. Monotone Schemesfor entropy convergence

30 / 79

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Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Monotone Scheme

Definition (Monotone schemes)

A scheme

un+1j = un

j −∆t

∆x(fj+ 1

2− fj− 1

2)

= unj −

∆t

∆x(f (un

j−p, · · · , unj+q)− f (un

j−p−1, · · · , unj+q−1))

= G (unj−p−1, · · · , un

j+q)

is called a monotone scheme if G is a monotonically nonde-creasing function G (↑, ↑, · · · , ↑) of each argument.

31 / 79

Comp. Math.&

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Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Monotone SchemeProposition

A first order conservative scheme

un+1j = un

j −∆t

∆x(f (uj , uj+1)− f (uj−1, uj ))

= G (unj−1, u

nj , u

nj+1).

is monotone, if f (↑, ↓) and ∆t∆x <

12L where L is the Lips-

chitz constant.

Proof.

∂G

∂unj−1

=∆t

∆x

∂ f

∂unj−1

≥ 0,∂G

∂unj+1

= −∆t

∆x

∂ f

∂unj+1

≥ 0,

∂G

∂unj

= 1− ∆t

∆x(∂ f

∂u−− ∂ f

∂u+) ≥ 1− ∆t

∆x2L > 0,

32 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Monotonicity of f

Monotonicity of f is a key assumption for a monotone scheme.

f (↑, ↓) :∂ f

∂u−≥ 0,

∂ f

∂u+≤ 0.

Intuitively, it respects the upwind principle in a mathematicalrigorous way.

• Lax-Friedrich flux is a monotone flux.

f (uj , uj+1) =1

2[f (uj ) + f (uj+1)− α(uj+1 − uj )]

∂ f

∂uj=

1

2[f ′(uj ) + α] ≥ 0,

∂ f

∂uj+1=

1

2[f ′(uj+1)− α] ≤ 0

for α = maxu|f ′(u)|.

33 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Roe flux is not monotone

Assume u− < 0, u+ > 0 and |u−| < |u+|,

f (u−, u+) = f (u−)⇒ ∂ f

∂u−=

∂f

∂u−< 0

which violates the monotonicity condition ∂ f∂u− ≥ 0.

34 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Good properties of monotoneschemes

• uj ≤ vj , ∀j implies G (u)j ≤ G (v)j , ∀j .• Local maximum principle:

mini∈stencils around j

ui ≤ G (u)j ≤ maxi∈stencils around j

ui

• L1−contraction:

‖G (u)− G (v)‖L1 ≤ ‖u − v‖L1 .

• Total variation diminishing (TVD) property:

‖G (u)‖BV ≤ ‖u‖BV .

where ‖u‖BV :=∑

j |uj+1 − uj |.

35 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Recall well-posedness properties

Proposition

• Bounded variation

V ba (u) := sup

∑i

|u(xi )− u(xi+1)| <∞, for any partition {xi}.

V R−R u(·, t) ≤ V R+st

−R+st u0(x), s = max |f ′(u)|.

• L1 contraction property. If u(x , t) and v(x , t) are solutions ofut + f (u)x = 0 with initial data u0(x) and v0(x).

‖u(x , t)− v(x , t)‖L1 ≤ ‖u0(x)− v0(x)‖L1 .

• L∞ maximum principle. If u(x , t) is a solution of scalarconservation laws with initial data u0(x), then ∀x , t

max u0(x) ≥ max u(x , t),

min u0(x) ≤ min u(x , t).

36 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Remark

• Properties of monotone scheme correspond towell-posedness properties of the PDE.

• For nonlinear problems such as HCL:stability is assessed in terms of TV norm.

37 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Entropy convergence

Theorem (Entropy convergence)

Solution to a monotone scheme satisfies all entropy con-ditions.

38 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Bottleneck

Theorem (Godunov)

Monotone schemes are at most first order accurate.

39 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Linear schemesDefinition (Linear schemes)

A scheme is called a ”linear scheme” if it is linear whenapplied to a linear PDE:

ut + aux = 0

where a is a constant.

A linear scheme forut + ux = 0

can be written as

un+1j =

l=k∑l=−k

cl (λ)unj+l

where cl (λ) are constants which may depend on λ = ∆t∆x . A linear

scheme is monotone iff

cl (λ) ≥ 0, ∀l .40 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Equivalence of monotone, TVDand monotonicity-preserving

Theorem

• Monotone ⇒ TVD.

• TVD ⇒ monotonicity-preserving.

• For linear schemes, monotonicity-preserving ⇒monotone.

Definition: Monotonicity preserving:

{unj+1 ≥ un

j ,∀j} ⇒ {un+1j+1 ≥ un+1

j , ∀j}

41 / 79

Comp. Math.&

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Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

TVD

Corollary

For linear schemes, monotonicity-preserving and TVDschemes are at most first order accurate.

42 / 79

Comp. Math.&

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Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Monotone schemes: summary

Given {uj}j

1 Evaluate numerical fluxes with monotonicity f (↑, ↓)

fj+ 12

= f (u−j+ 1

2

, u+j+ 1

2

)first order

= f (uj , uj+1), ∀j .


un+1j = un

j −∆t

∆x

(fj+ 1

2− fj− 1

2

).

Entropy convergence gauranteed!

43 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Conservative schemes: summary

Given {uj}j

1 Evaluate numerical fluxes intuitively with upwind principle

fj+ 12

= f (u−j+ 1

2

, u+j+ 1

2

)first order

= f (uj , uj+1), ∀j .


un+1j = un

j −∆t

∆x

(fj+ 1

2− fj− 1

2

).

Only weak convergence gauranteed; entropy convergencemay not hold!

44 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Road Map






45 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

4. Nonlinear TVD Schemes

46 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

General framework of finite volumescheme

A finite volume scheme is of the form

d

dtuj +

1

∆xj

[fj+ 1

2− fj− 1

2

]= 0

where fj+ 12

is the numerical flux. We want

fj+ 12≈ f (u(xj+ 1

2, t)).

For the time being, assume f ′(u) ≥ 0. Then, based on theupwind principle

fj+ 12

= f (u−j+ 1

2

).

47 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Spatial approximation: first order

d

dtuj +

1

∆xj

[f (u−

j+ 12

)− f (u−j− 1

2

)

]= 0

First order approximation,

u−j+ 1

2

= u(x−j+ 1

2

) ≈ uj + O(∆x).

Thend

dtuj +

1

∆xj[f (uj )− f (uj−1)] = 0

Then, with a first order time discretization, the schemebecomes

un+1j = un

j −∆t

∆x

(f (un

j )− f (unj−1)

).

48 / 79

Comp. Math.&

Applications

Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

Second order spatial approximation

d

dtuj +

1

∆xj

[f (u−

j+ 12

)− f (u−j− 1

2

)

]= 0

• How to construct a second order approximation to u−j+ 1

2

,

given cell averages {uj}j∈S in the neighborhood?

• The procedure of going from cell averages {uj} to pointvalues at cell boundaries u−

j+ 12

is called a reconstruction.

• Note the difference between reconstruction and traditionalinterpolation.

Recall for the first order scheme S = Ij . For the second orderscheme,

• S = {Ij−1, Ij}.• S = {Ij , Ij+1}.

49 / 79

Comp. Math.&

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Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

S = {Ij−1, Ij}

1 Choose Stencil {Ij−1, Ij} :

2 Construct a polynomial P1 = a + bx such that

1

∆x

∫Ij−l

P1(x) dx = unj−l , l = 0, 1

P1(x) = unj +

unj − un

j−1

∆x(x − xj )

3 Evaluate the polynomial at cell interface

u(1)

j+ 12

= P1(x−j+ 1

2

) = −1

2un

j−1 +3

2un

j ≈ u(x−j+ 1

2

) + O(∆x2).

50 / 79

Comp. Math.&

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Jingmei Qiu

Outline

Nonlinearequations

Conservativeschemes

Monotoneschemes


Summary

S = {Ij , Ij+1}

1 Choose Stencil {Ij , Ij+1} :

2 Construct a polynomial P1 = a + bx such that

1

∆x

∫Ij−l

P1(x) dx = unj+l , l = 0, 1

P1(x) = unj +

unj+1 − un

j

∆x(x − xj )

3 Evaluate the polynomial at cell interface

u(2)

j+ 12

= P1(x−j+ 1

2

) =1

2un

j +1

2un

j+1 ≈ u(x−j+ 1

2

) + O(∆x2).

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Summary

Recalld

dtuj +

1

∆xj

[f (u−

j+ 12

)− f (u−j− 1

2

)

]= 0

• u−j+ 1

2

is approximated by a reconstruction.

• Depending on the stencil, u(1)

j+ 12

and u(2)

j+ 12

provide second

order approximations, while uj is a first orderapproximation.

• Which one to choose is from the consideration of stability,in terms of the TV norm (to be discussed next).

• u−j− 1

2

can be done similarly.

• Time discretization can be done by first order forwardEuler or explicit TVD (SSP) Runge-Kutta methods, TBD.

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Summary

Schemes with TV stability

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Summary

Minmod function

1 Let

u(1)j = u

(1)

j+ 12

− unj =

1

2(un

j − unj−1),

u(2)j = u

(2)

j+ 12

− unj =

1

2(un

j+1 − unj ).

2 Define

minmod(a, b) =

{sign(a) min(|a|, |b|), a · b ≥ 0

0, a · b < 0.

3 Consider

fj+ 12

= f (unj + un

j ) = f (unj + minmod(u

(1)j , u

(2)j ))

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Summary

MUSCL

MUSCL: Monotone Upstream Scheme for Conservation Laws

un+1j = un

j −∆t

∆x

(fj+ 1

2− fj− 1

2

)with

fj+ 12

= f (unj + un

j ) = f (unj + minmod(u

(1)j , u

(2)j ))

fj− 12

= f (unj−1 + un

j−1) = f (unj−1 + minmod(u

(1)j−1, u

(2)j−1))

• MUSCL scheme is a nonlinear scheme, as the schemedepends locally on the solution u.

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Summary

MUSCL is TVD

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Proposition (MUSCL is TVD)

MUSCL scheme is TVD, TV (un+1) ≤ TV (un) i.e.∑j

|un+1j+1 − un+1

j | ≤∑

j

|unj+1 − un

j |

Proof. By using Harten’s TVD Lemma.

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Summary

Harten’s TVD Lemma

Lemma (Harten)

If a scheme can be written as

un+1j+1 = un

j + λ(Cj+ 1

2∆+u

nj − Dj− 1

2∆−u

nj

)with

Cj+ 12≥ 0, Dj− 1

2≥ 0, 1− λ(Cj+ 1

2+ Dj+ 1

2) ≥ 0,

with λ = ∆t∆x , then the scheme is TVD. Here

∆+uj = uj+1 − uj , ∆−uj = uj − uj−1.

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Summary

MUSCL scheme and TVD propertyTo prove that MUSCL scheme enjoys the TVD property, it issufficient to check assumptions of Harten’s TVD Lemma.MUSCL scheme can be written as

un+1j = un

j −∆t

∆x

(f (un

j + unj )− f (un

j−1 + unj−1)

)= un

j − λ[Dj− 1

2∆−u

nj

]• Cj− 1

2= 0.

• Dj− 12∈ [0, 3

2 maxu |f ′(u)|].

Dj− 12

=f (un

j + unj )− f (un

j−1 + unj−1)

unj − un

j−1

= f ′(ξ)un

j − unj−1 + un

j − unj−1

unj − un

j−1

= f ′(ξ)

1 +un

j

unj − un

j−1︸︷︷︸0≤·≤ 1

2

−un

j−1

unj − un

j−1︸︷︷︸0≤·≤ 1

2

.

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MUSCL scheme and TVD property

To prove that MUSCL scheme enjoys the TVD property, it issufficient to check assumptions of Harten’s TVD Lemma.MUSCL scheme can be written as

un+1j = un

j −∆t

∆x

(f (un

j + unj )− f (un

j−1 + unj−1)

)= un

j − λ[Dj− 1

2∆−u

nj

]

• Cj− 12

= 0.

• Dj− 12∈ [0, 3

2 maxu |f ′(u)|].• 1− λ(Cj− 1

2+ Dj− 1

2) ≥ 0.

If λmax |f ′(ξ)| ≤ 23 , then 1− λDj− 1

2≥ 0.

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Summary

MUSCL coupled with second order RK

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Second order Runge-Kutta method

Let F (u)j = − 1∆x (fj+ 1

2− fj− 1

2).

1 Stage 1:

u(1) = un + ∆tF (un):= L(un)

2 Stage 2:

un+1 = un +1

2∆t(F (un) + F (u(1))

)=

1

2un +

1

2u(1) +

1

2∆tF (u(1))

=1

2un +

1

2L(u(1))

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Summary

TVD Runge-Kutta method2nd order Runge-Kutta method can be written as:

u(1) = L(un),

un+1 =1

2(un + L(u(1))).

Then,

TV (u(1)) ≤ TV (un)

TV (un+1) ≤ 1

2TV (un) +

1

2TV (L(u(1)))

≤ 1

2TV (un) +

1

2TV (u(1))

≤ 1

2TV (un) +

1

2TV (un)

≤ TV (un)

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Summary

TVD (SSP) RK methods

• If the scheme enjoys the TVD stability with the forwardEuler time discretization, this second order RK methodalso enjoys the TVD stability. Thus, it is call the TVD orstrong stability preserving (SSP) RK method.

• The idea is to rewrite the RK method as a convexcombination of forward Euler methods.

• Such concept can be generalized to a third order TVD orSSP RK method.

• There does not exist fourth order or higher order TVD orSSP RK methods under the convex combinationassumption. 3

3Strong Stability-Preserving High-Order Time Discretization Methods,2001, SIAM Review, Sigal Gottlieb, Chi-Wang Shu, and Eitan Tadmor

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Summary

Third order TVD (SSP) RKmethods

u(1) = un + ∆tF (un)

u(2) = un + ∆t

(1

4F (un) +

1

4F (u(1))

)un+1 = un + ∆t

(1

6F (un) +

2

3F (u(2)) +

1

6F (u(1))

)

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Proposition

If TV (u(1)) ≤ TV (un), then for the third order TVD RKmethod, we have

TV (un+1) ≤ TV (un).

Proof. First

u(1) = L(un)⇒ TV (u(1)) ≤ TV (un)

Then from u(2) = 34 u

n + 14L(u(1)), we have

TV (u(2)) ≤ 3

4TV (un) +

1

4TV (u(1)) ≤ TV (un).

Finally from un+1 = 13 u

n + 23L(u(2)), we have

TV (un+1) ≤ 1

3TV (un) +

2

3TV (u(2)) ≤ TV (un).

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Summary

MUSCL: generalization.

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MUSCL scheme: f ′(u) change sign

General case: f ′(u) changes sign

d

dtuj +

1

∆xj

[fj+ 1

2− fj− 1

2

]= 0

where fj+ 12

= f (u−j+ 1

2

, u+j+ 1

2

), ∀j .

• f (↑, ↓): respecting the upwind principle— e.g. Lax-Friedrich flux.

• u−j+ 1

2

reconstructed from a stencil including Ij .

— e.g. {Ij−1, Ij}, or {Ij , Ij+1} or with minmod.

• u+j+ 1

2

reconstructed from a stencil including Ij+1

— e.g. {Ij , Ij+1}, or {Ij+1, Ij+2} or with minmod.

67 / 79

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Nonlinearequations

Conservativeschemes

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Summary

MUSCL scheme: high order RK

• TVD stability can be proved by checking assumptions inthe Harten’s TVD Lemma.

• The TVD (SSP) RK can be applied to ensure

TV (un+1) ≤ TV (un).

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Nonlinearequations

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Summary

Generalized MUSCL: higher orderreconstruction

• Stencil {Ij−2, Ij−1, Ij} : find P2 polynomial such that

1

∆x

∫Ij−l

P2(x) dx = uj−l , l = 0, 1, 2.

Then,

u(1)

j+ 12

= P2(xj+ 12) =

1

3uj−2 −

7

6uj−1 +

11

6uj .

• Stencil {Ij−1, Ij , Ij+1} :

u(2)

j+ 12

= −1

6uj−1 +

5

6uj +

1

3uj+1.

• Stencil {Ij , Ij+1, Ij+2, } :

u(3)

j+ 12

=1

3uj +

5

6uj+1 −

1

6uj+2.

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Summary

Generalized MUSCL

Define

u−j+ 1

2

= u(2)

j+ 12

− uj , u+j+ 1

2

= uj+1 − u(3)

j+ 12

.

Then,

u−j+ 1

2

= uj + minmod(u−j+ 1

2

, uj+1 − uj , uj − uj−1)

u+j+ 1

2

= uj+1 −minmod(u+j+ 1

2

, uj+1 − uj , uj+2 − uj+1)

which are used to evaluate the flux fj+ 12

= f (u−j+ 1

2

, u+j+ 1

2

) in a

conservative scheme.

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Full accuracy in monotone regions

Proposition

In smooth and monotone regions, the generalized MUSCLscheme maintains its original high order accuracy fromreconstruction.

Proof.

• u−j+ 1

2

= u(2)

j+ 12

− uj = 12ux ∆x +O(∆x2)

• uj+1 − uj = ux ∆x +O(∆x2)

• uj − uj−1 = ux ∆x +O(∆x2)

In a monotone region, with mesh refinement, these threequantities are all of the same sign; and the magnitude of thefirst one is roughly half of the latter ones.

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Accuracy degeneration nearextrema

Theorem (Osher)

TVD schemes are at most first-order accurate nearsmooth extrema.

Observation: the minmod limiter is always being activatedaround extrema.

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Summary

Summary of MUSCL

• Stability:— , TVD stability, when coupled with second or thirdorder TVD (SSP) RK method.

TV (u(n+1)) ≤ TV (un).

• Accuracy:— , high order accuracy in smooth and monotoneregions.— / first order accuracy around extrema. (motivation forTVB schemes)

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Road Map






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5. Nonlinear TVB Schemes

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Modified minmod limiter

u−j+ 1

2

= uj + minmod(u−j+ 1

2

, uj+1 − uj , uj − uj−1)

u+j+ 1

2

= uj+1 −minmod(u+j+ 1

2

, uj+1 − uj , uj+2 − uj+1)

with

minmod(a, b, c) =

{a, if |a| ≤ M∆x2

minmod(a, b, c), otherwise..

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Summary

TVB

Total variation may be increase only around finite number ofextrema.

TVB : TV (un+1) ≤ TV (un) + C∆t.

HenceTV (un) ≤ C (T ), for n∆t ≤ T .

• , The modified minmod limiter de-activates the minmodlimiter around smooth extrema.

• / M is a parameter related to the second derivative of u,and has to be tuned case by case.

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Summary

• Conservative scheme:

d

dtuj +

1

∆xj

[fj+ 1

2− fj− 1

2

]= 0

Mass conservation; convergence to the weak solution.

• Monotone scheme:f (u−, u+), as approximations to a local Riemann problem,is monotone (respect the upwind principle).convergence to the entropy solution

• TVD and TVB scheme:reconstruction of u±

j+ 12

with high order reconstruction

procedures, yet maintain TVD or TVB stability.strike a balance between accuracy and stability

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Summary

Reference:

• B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor(Editor: A. Quarteroni), Advanced NumericalApproximation of Nonlinear Hyperbolic Equations, LectureNotes in Mathematics, volume 1697, Springer, 1998.

• C.-W. Shu, High Order Weighted EssentiallyNonoscillatory Schemes for Convection DominatedProblems, SIAM Rev., 51(1), 82-126.

Acknowledgement:

• Lecture notes of applied math course 257 by ProfessorChi-Wang Shu, when I was a graduate student at BrownUniversity.

• Special thanks to Ms. Mingchang Ding (Ph.D. student atUniversity of Delaware) for her help in preparing theseslides.

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topic course on numerical methods in computational fluid

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