WPPII Computational Fluid Dynamics I - Unicampphoenics/SITE_PHOENICS/Apostilas/CFD-1_U... · WPPII Computational Fluid Dynamics I In CFD, however, ... Initial data, conservative methods

Computational Fluid Dynamics IPPPPIIIIWWWW

Numerical Methods forWave Equation:

Part II: Discontinuous Solutions

Instructor: Hong G. ImUniversity of Michigan

Fall 2001


Part I• Method of Characteristics• Finite Volume Approach and Conservative Forms• Methods for Continuous Solutions

- Central and Upwind Difference- Stability, CFL Condition- Various Stable Methods

Part II• Methods for Discontinuous Solutions

- Burgers Equation and Shock Formation- Entropy Condition- Various Numerical Schemes

Outline

Solution Methods for Wave Equation


Exact Solutions


0=∂∂+

∂∂

xfc

tf

The analytic solution is obtained by characteristics

;0; ==dtdfc

dtdx

Consider 1st Order Wave Equation

t

f

f

xDiscontinuity of solution is allowed

(Weak Solution)

( )RLR

L ffxxfxxf

xf >

><

=0

0)0,(

cdtdx =


( )RLR

L ffxxfxxf

xf >

><

=0

0)0,(0=∂∂+

∂∂

xff

tf

;0; ==dtdff

dtdx

Inviscid Burgers’ Equation

t

x

Characteristics

0x

Lf

Rf

??=dtdx

0x

Shock Wave


( ) ( ) axa

axfaxffff

axf

xf

R

RLRL

L

<<−

>

−−+

−<

=21)0,(

A slight variation of the initial condition – formation of shock

t

x

Shock Wave

a− a

Lf

Rf 0=t

RL ffat−

= 2


( )RLR

L ffxxfxxf

xf <

><

=0

0)0,(0=∂∂+

∂∂

xff

tf

;0; ==dtdff

dtdx

Reverse Shock (?)

Characteristics

0x

Lf

Rf t

x0x

Unstable, entropy-violatingsolution


( )RLR

L ffxxfxxf

xf <

><

=0

0)0,(0=∂∂+

∂∂

xff

tf

;0; ==dtdff

dtdx

Rarefaction Wave (physically correct solution)

Characteristics

0x

Lf

Rf t

x0x

Computational Fluid Dynamics IPPPPIIIIWWWWWeak solutions to hyperbolic equations may not be unique.

How can we find a physical solution out of many weaksolutions?

In fluid mechanics, the actual physics always includes dissipation, i.e. in the form of viscous Burgers’ equation:

2

2

xf

xff

tf

∂∂=

∂∂+

∂∂ ε

Therefore, what we are truly seeking is the solution to the viscous Burgers’ equation in the limit of 0→ε

Computational Fluid Dynamics IPPPPIIIIWWWWIn CFD, however, we do not want to solve the viscousBurgers’ equation with extremely small because it becomes computationally expensive.

Nevertheless, in many gas dynamics application, thereis need to capture discontinuous solution behavior.

And we want to reproduce the weak solution behaviornumerically by solving inviscid Burgers’ equation.

ε

- Shock speed- Entropy condition- Wave shape (without smearing or wiggles)


Finite Difference Methodsfor

Discontinuous Solutions

Computational Fluid Dynamics IPPPPIIIIWWWWShock Speed for the Burgers’ Equation (Inviscid)

In conservative form,

2

21,0 fF

xF

tf ==

∂∂+

∂∂

LL Ff ,RR Ff ,

C

CV (Fixed)

Net flux: RL FF −

Rate of change within CV:

( )Cff RL −

Shock Speed - 1

Computational Fluid Dynamics IPPPPIIIIWWWWRate of Change = Net Flux

( ) RLRL FFCff −=−

RL

RL

ffFFC

−−=

Since 2

21 fF =

( )( ) ( )RLRL

RLRL

RL

RL ffff

ffffffffC +=

−+−=

−−=

21

21

21 22

Shock Speed

Shock Speed - 2


In capturing the correct solution behavior for discontinuousInitial data, conservative methods are essential.

Conservative Methods for Nonlinear Problems

Example: inviscid Burgers equation:

021or0 2 =

∂∂+

∂∂=

∂∂+

∂∂ f

xtf

xff

tf

Conservative Method - 1


Consider upwind and forward Euler scheme:

Example: for inviscid Burgers equationwith discontinuous initial data

0=+ xt fff

1−j

1

j 1+j

( ) 011 =−∆−= −+ n

jnj

nj

nj

nj fff

htff

Non-conservative form

Never moves!

( ) 0221 =+

xt ff

( )htff

htff n

jnj

nj

nj 2

)()(2

21

21 ∆=−∆−= −+

Conservative form

Conservative Method - 2


Entropy Condition:A discontinuity propagating with speed C satisfies the entropy condition if

For a conservation equation

)()( RL fFCfF ′>>′

( ) ( ) ( ) ( )R

R

L

L

fffFfFC

fffFfF

−−≥≥

−−

Entropy Condition - 1

0)( =∂

∂+∂∂

xfF

tf

Version I

Version II

And some others…


Examples: Godunov methodApproximate Riemann solverArtificial dissipation

In practice, rather than applying a separate entropy condition to the numerical solution procedure, numericalmethods are developed first and then it can be provedthat the methods satisfy the entropy condition.

Entropy Condition - 2

Computational Fluid Dynamics IPPPPIIIIWWWWExample Problem: Linear Wave Equation

0,,0 ≥∞<<∞−=∂∂+

∂∂ tx

xfU

tf

><

=0001

)0,(xx

xf

Exact Solution: )( Utxf −

Apply various numerical methods discussed in Lecture 8 to the problem.


Stable forBeam-Warming

Stable forLax-Wendroff I

Conditionally consistent

Stable for

Lax-Friedrichs

Stable forUpwind

0=+ xt Uff

1≤λ

( )

( ) 022

243

212

2

211

=+−∆−

+−+

∆−

−−

−−+

nj

nj

nj

nj

nj

nj

nj

nj

fffh

tUh

fffU

tff

20 ≤≤ λ

011

=−

+∆− −

+

hff

Ut

ff nj

nj

nj

nj

( )

( ) xxx

xx

fUh

fUh

1326

12

22

+−−

−

λλ

λ

1≤λ

( ) xxx

xx

fUh

fUh

22

13

12

λ

λλ

−+

−

( )

( )0

2

2/

11

111

=−

+

∆+−

−+

−++

hff

U

tfff

nj

nj

nj

nj

nj

( )

( ) xxxx

xxx

fUh

fUh

23

22

18

16

λλ

λ

−−

−−( )

( )0

22

2

2

1122

111

=+−

∆−

−+

∆−

−+

−++

hfff

tU

hff

Ut

ff

nj

nj

nj

nj

nj

nj

nj

1≤λ

( )( )

( ) ( ) xxxx

xxx

fUh

fUh

λλ

λλ

−−−

−−

218

216

23

2

Computational Fluid Dynamics IPPPPIIIIWWWWComparison: 5.0at;01.0 == th

LeVeque, R. J., Numerical Methods for Conservation Laws, Birkhäuser, 1992

Computational Fluid Dynamics IPPPPIIIIWWWWComparison: 5.0at;0025.0 == th

LeVeque, R. J., Numerical Methods for Conservation Laws, Birkhäuser, 1992

Computational Fluid Dynamics IPPPPIIIIWWWWFirst-Order Methods and Diffusion

Upwind:

( ) ( ) xxxxx fUhfUhxfU

tf 132

61

22

2

+−−−=∂∂+

∂∂ λλλ

Lax-Friedrichs:

( ) xxxxx fUhfUhxfU

tf 2

2

13

12

λλλ

−+

−=

∂∂+

∂∂

Dissipative = Smearing

Computational Fluid Dynamics IPPPPIIIIWWWWSecond-Order Methods and Dispersion

Lax-Wendroff:

( ) ( ) xxxxxxx fUhfUhxfU

tf 2

32

2

18

12

λλλ −−−−=∂∂+

∂∂

Beam-Warming:

( )( ) ( ) ( ) xxxxxxx fUhfUhxfU

tf λλλλ −−−−−=

∂∂+

∂∂ 21

821

62

32

Dispersive = Wiggles


Observation 1:Second-order methods tends to capture shaper solution (better accuracy), but they produce wiggly solutions.

Observation 2:First-order methods are dissipative and less accurate, but the solution does not oscillate. (preserves monotonicity).


Question: How can we avoid oscillatory behavior ?

(How can we achieve monotone behavior ?)

Godunov Theorem (1959):

“Monotone behavior of a numerical solution cannot be assured for finite-difference methods with more thanfirst-order accuracy.”

Computational Fluid Dynamics IPPPPIIIIWWWW Godunov Theorem - 1

0=∂∂+

∂∂

xfU

tf

For linear equation

which yields:2

22

2

2

;x

fUtf

xfU

tf

∂∂=

∂∂

∂∂−=

∂∂

A general linear scheme can be written as:

∑ ++ =

k

nkjk

nj fcf 1

Expanding in Taylor series aroundnkjf +

njf

)(2

32

222

hOx

fhkxfkhff n

jn

kj +∂∂+

∂∂+=+


Substituting

Also, can be Taylor expanded in time around

∑ ∑∑ +∂∂+

∂∂+=+

k kk

kkk

nj

nj hOck

xfhkc

xfhcff )(

232

2

221

(a)=(b) (2nd order accurate):2

2;;1

∆=∆−== ∑∑∑ h

tUckh

tUkcck

kk

kk

k

1+njf n

jf

)(2

32

221 tO

tft

tftff n

jnj ∆+

∂∂∆+

∂∂∆+=+

)(2

32

222 tO

xftU

xftUf n

j ∆+∂∂∆+

∂∂∆−=

(a)

(b)


Condition for monotonicity:11

11 then,If ++++ >> n

jnj

nj

nj ffff

Since ∑ ++ =

k

nkjk

nj fcf 1

( )∑ +++++

+ −=−k

nkj

nkjk

nj

nj ffcff 1

111

> 0The above should be valid for arbitrary

00 111 >−⇔> +++

nj

njk ffc

njf

Is it possible?


Hence we get:

22;;1

∆=∆−== ∑∑∑ h

tUckh

tUkcck

kk

kk

k

22

=

∑∑∑

kk

kk

kk kcckc

22222

=

∑∑∑

kk

kk

kk keeke

Define: )0(2 >= kkk cec

This violates Cauchy inequality ( )( ) ( )2babbaa ⋅≥⋅⋅

Monotone 2nd Order scheme is impossible!


The Godunov theorem suggests that monotone schemes are limited to the first order schemes.

However, Godunov theorem is valid only for linear schemes.

The Godunov theorem can be circumvented by nonlinear schemes, even for linear equations.

⇒ Basis for higher-order monotone schemes(TVD, higher-order upwind)


In the following, we will discuss various approaches to achieve monotone solution for Euler equations:

- Flux-splitting- Artificial viscosity- Godunov method- Higher-order upwind and TVD

Computational Fluid Dynamics IPPPPIIIIWWWWGeneralized Upwind Scheme (for both U > 0 and U < 0 )

0),( 11 >−∆−= −+ Uff

htUff n

jnj

nj

nj

0),( 11 <−∆−= ++ Uff

htUff n

jnj

nj

nj

Define( ) ( )UUUUUU −=+= −+

21,

21

The two cases can be combined into a single expression:

[ ])()( 111 n

jnj

nj

nj

nj

nj ffUffU

htff −+−∆−= +

−−

++

Upwind Scheme - Revisited

Flux Splitting


Or, substituting

Recall Lax-Wendroff

)2(2

)(2 1111

1 nj

nj

nj

nj

nj

nj

nj fff

htU

ffhtUff −+−+

+ +−∆

+−∆−=

−+ UU ,

central difference + artificial viscosity

( ) ( )nj

nj

nj

nj

nj

nj

nj fff

htUff

htUff 112

22

111 2

22 −+−++ +−∆+−∆−=



Two options for hyperbolic equations:

1. Use directionally-biased upwind scheme(Flux splitting)

2. Use central differencing and introduce numericalviscosity (Artificial viscosity)


Computational Fluid Dynamics IPPPPIIIIWWWWSteger-Warming (1979)

A system of hyperbolic equations

Flux Splitting - 1

0=∂∂+

∂∂

xtEU

can be written in the form

[ ] [ ]UEUU∂∂==

∂∂+

∂∂ A

xA

t;0

The system is hyperbolic if

[ ] [ ][ ] [ ] [ ]

== −−

T

T1

11 ;

N

TTATq

q!λ

Computational Fluid Dynamics IPPPPIIIIWWWWFlux Splitting - 2

[ ] [ ][ ][ ] UUE 1−== TTA λ

The matrix of eigenvalues is divided into two matrices

[ ] [ ] [ ] [ ][ ][ ] [ ][ ][ ] 11 −−−+−+ +=+= TTTTAAA λλHence

[ ]λ[ ] [ ] [ ]−+ += λλλ

Define −+ += EEE [ ] [ ]( )UEUE −−++ == AA ,

Conservation law becomes

0=∂∂+

∂∂+

∂∂ −+

xxtEEU


For example, first-order explicit upwind scheme

( ) ( )njjn

jjnj

nj h

tht −−

++−

++ −∆−−∆−= EEEEUU 111

How to split?

( ) ( )[ ]njjjjnj h

t −+−

−+

+ +−+∆−= EEEEU 11


Example: 1-D Hyperbolic Equation

02

22

2

2

=∂∂−

∂∂

xfc

tf

xfw

tfv

∂∂=

∂∂= ;0

010 2

=

−

−+

x

x

t

t

wvc

wv

Leading to

[ ]

−

−=

∂∂=

−−

=

=

010

;;22 c

Av

wcwv

UEEU


Leading to

[ ]

−=

=−

cc

T11

T2

T11

qq [ ] [ ]

−

=

= −+

cc

000

;000

λλ

−−−−

=

+−−

= −+

cwvwccv

cwvwccv 22

21;

21 EE

[ ] [ ][ ][ ] [ ] [ ][ ][ ] 11; −−−−++ == TTATTA λλ

[ ] [ ]UEUE −−++ == AA ;

Computational Fluid Dynamics IPPPPIIIIWWWWArtificial Viscosity [Ref: Hirsch, vol. 2, Ch. 17]

The centered difference method (e.g. Lax-Wendroff) - Second order accuracy- Oscillation near discontinuity

In order to “damp out” oscillation, we can either- Use implicit numerical viscosity (upwind) or- Add an explicit numerical viscosity

Artificial Viscosity - 1

Computational Fluid Dynamics IPPPPIIIIWWWWVon Neumann and Richtmyer (1950)

0=∂∂+

∂∂

xF

tf

xfDh

xfFF

∂∂=

∂∂−=′ 2; αα

- Simulates the effect of the physical viscosity on thegrid scale, concentrated around discontinuity and negligible elsewhere.

- h2 is necessary to keep the viscous term of higher order.


Computational Fluid Dynamics IPPPPIIIIWWWWFor the fluid-dynamic system of equations (Euler equations):

0)/(

2 =

++

∂∂+

∂∂

ρρρρ

ρρρ

pEupu

u

xEu

t

epueE ργ )1(;2/2 −=+=Add the artificial viscosity to RHS:

where

∂∂

∂∂

∂∂=

xu

xu

uh

x10

2ρα Project #2


Computational Fluid Dynamics IPPPPIIIIWWWWGodunov Method – FVM + Riemann solver

[ ] 0)( =∂∂+

∂∂ fF

xtf

Consider a 1-D Conservation Eq:

Integrating across the j-th cell:

Godunov Method - 1

Fj-1/2 Fj+1/2

1+jff j−1

jf

xj j+1j−1

0)()( 2/12/1

2/1

2/1=−+

∂∂

−+∫+

−jj

x

xxFxFfdx

tj

j

Define the cell-average: ( )2/12/1/2/1

2/1−+ −= ∫

+

−jj

x

xav xxfdxf j

j

( ) 0)()( 2/12/12/12/1 =−+∂∂− −+−+ jj

avxj xFxF

tfxx

Computational Fluid Dynamics IPPPPIIIIWWWWFor simplicity, assuming uniform grid,

Integrating in time by

where the time integration is done by solving Riemann Problem for each cell boundary:

Godunov Method - 2

∆−

∆∆−= ∫ ∫

∆+ ∆+

−++

tt

t

tt

t jjn

avn

av dtFt

dtFth

tff 2/12/11 11

hxx xj =− −+ 2/12/1

t∆

[ ] 0)( =∂∂+

∂∂ fF

xtf

>≤

=++

+

2/11

2/1)0,(jj

jj

xxfxxf

xf 1+jfjf

xj j+1

t

2/1+= jcdtdx

t+∆t

Computational Fluid Dynamics IPPPPIIIIWWWWWave Diagram:

Godunov Method - 3

1+jff j−1jf

j j+1j−1

ShockShockRarefaction

Wave

Computational Fluid Dynamics IPPPPIIIIWWWWGodunov Method: The Procedure

Godunov Method - 4

1. Construct a cell average value

( )2/12/1/2/1

2/1−+ −= ∫

+

−jj

x

xav xxfdxf j

j

2. Solve a Riemann problem to find the time integration of fluxes

3. Construct new

4. Go to 1.

avf

ttt ∆+=

Computational Fluid Dynamics IPPPPIIIIWWWWExample: Inviscid Burgers Equation

Godunov Method - 5

021 2 =

∂∂+

∂∂ f

xtf

>≤

=++

+

2/11

2/1)0,(jj

jj

xxfxxf

xf

Initial Condition:

Characteristic Velocity:

21

2/12/1

+

++

+=

= jj

jj

ffdtdxc

Computational Fluid Dynamics IPPPPIIIIWWWWThe Formula:

Godunov Method - 6

1+> jj ff

><

=++

+

2/11

2/1

//

jj

jj

ctxfctxf

f

Shock waves

Expansion Waves

( ) 2/12/1 ++ += jjj ffc

<

>=

++

+

+ 0

0

2/12

121

2/12

21

2/1

jj

jj

j cf

cfF

1+< jj ff

><<

<=

++

+

+

2/11

1

2/1

///

/

jj

jj

jj

ftxfftxftx

ftxff

<<<

>>>

<<

=

+++

++

+

+

00

00

00

12/12

121

12/12

21

1

2/1

jjjj

jjjj

jj

j

ffcf

ffcf

ff

F

Computational Fluid Dynamics IPPPPIIIIWWWWGraphical Representation of the Procedure

Godunov Method - 7

njavf 1, +

njavf 1, −

njavf ,

j j+1j−1

2

3. t = t+∆t

1,+n

javf1

1,++

njavf1

1,+−

njavf

t = t+∆t

1

t = t+∆t3

t = t


Godunov method implicitly assumes that the waves fromadjacent cells do not interact.

⇒ The wave can travel at most half cell in distance during one time step

Godunov Method - 8

21

max ≤∆htf

Stability Condition

Computational Fluid Dynamics IPPPPIIIIWWWWSpecial Case: Linear Advection Equation

Godunov Method - 9

0,0 >=∂∂+

∂∂ U

xfU

tf

for which the Riemann problem is trivial and we get

( )nj

nj

njav

njav ff

htUff 1,

1, −+ −∆−=

jj UfF =+ 2/1

: 1st Order Upwind Scheme

Computational Fluid Dynamics IPPPPIIIIWWWWFor general nonlinear problems, the Riemann problemrequires nonlinear iterations and becomes expensive.

Godunov Method - 10

Approximate Riemann Solver : Flux determined from local conditions.

- Roe Scheme: allows expansion shock (nonphysical)

- Enquist-Osher scheme: removing expansion shock(entropy condition)

Computational Fluid Dynamics IPPPPIIIIWWWWGodunov method can be expressed as

Approximate Riemann Solver - 1

where

[ ]),(),( 111 n

jnj

nj

nj

nj

nj ffff

htff −+

+ Φ−Φ∆−=

( )),(),( 1*

1nj

nj

nj

nj fffFff ++ =Φ

and is the exact Riemann solution at ),( 1* n

jnj fff + 2/1+jx

Instead of solving for nonlinear Riemann solution,Roe suggested an approximate Riemann solution

),(ˆ1

* nj

nj fff + which depends on the local conditions

Computational Fluid Dynamics IPPPPIIIIWWWWFor example, in solving the Riemann problem for Burgers equation, instead of

Approximate Riemann Solver - 2

Solve

021 2 =

∂∂+

∂∂ f

xtf

where is a local value of the solution at

2ˆˆ 1

1

12/1

+

+

++

+=

−−

== jj

jj

jjj

ffffFF

ff

0ˆ =∂∂+

∂∂

xff

tf

f̂ 2/1+j

which allows expansion shock – fixed by Enquist & Osher

Computational Fluid Dynamics IPPPPIIIIWWWWHigher Order Upwind Method:

MUSCL (Monotone Upstream-centered Schemesfor Conservation Laws) - Van Leer (1979)

Higher Order Upwind

Godunov method starts from the cell average values,and solve Riemann problem.

To increase the order of accuracy, construct a piecewiselinear approximation within in each cell in determining theflux terms.

Coupled with approximate Riemann solver.

Computational Fluid Dynamics IPPPPIIIIWWWWHigher Order Upwind

j j+1j−1 j+2

1+jf 2+jf

jf1−jf

Ljf 2/1+

Rjf 2/1+

( ) ( )jjjjjLj ffffff −++−−+= +−+ 112/1 4

14

1 κκ

( ) ( )12112/1 41

41

+++++ −−−−−−= jjjjjRj ffffff κκ

−

=central1Fromm0upwind1

κ

1for −=κ

Computational Fluid Dynamics IPPPPIIIIWWWWTotal Variation Diminishing (TVD) Scheme

Higher order upwind scheme improves accuracy, but itstill generates non-monotone behavior

TVD scheme can achieve higher order accuracy and monotonicity

∫ ∂∂= dx

xfTV

Harten (1983):

Monotone scheme ⇔ TVD

⇒⇒⇒⇒ Aero 623

Documents

WPPII Computational Fluid Dynamics I - Unicampphoenics/SITE_PHOENICS/Apostilas/CFD-1_U... · WPPII Computational Fluid Dynamics I In CFD, however, ... Initial data, conservative methods