CFD03 Flux Functions

8/12/2019 CFD03 Flux Functions

1/13

26.03.2004

Sl ide 1 Hchstleistungsrechenzentrum Stut tgar t

C.-D. Munz1, S. Roller2, M. Dumbser1

University of Stuttgart1Institute for Aerodynamics and Gas Dynamics (IAG)

www.iag.uni-stuttgart.de2High-Performance Computing-Center Stuttgart (HLRS)

www.hlrs.de

Introduction to Computational Fluid Dynamics

3

Flux Functions

2

Contents

1. Equations

2. Finite Volume Schemes

3. Linear Advection Equation

4. Systems of Advection Equations

5. Scalar Conservation Law

6. One-dimensional Euler Equations

7. Godunov-Type Schemes

8. Flux-Vector Splitting Schemes

9. Second Order Accuracy MUSCL Schemes

10. Boundary Conditions

11. Finite-Volume Schemes in Multi-Dimensions

12. ENO-/ WENO Schemes

13. Discontinuous Galerkin Finite Element Methods


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3

7. Godunov Type Schemes

Preliminaries

Simplest flux calculation

( ) ( ) ( )( )112

1,

21 +++

+== iiiii ufufuugg

only stable by adding artifical viscosity

xxxxxx uxux42

,

Jameson scheme for transonic flow

Runge-Kutta time step schemesophisticated choice of dissipation

4

Piecewise constant solution

Solve the local Riemann problems exactly

Average the exact solution

Godunovs Idea

x

u

xi-1/2 xi+1/2 xi+3/2

n

iu

n

2iu

+n

1iu

n

1iu

+


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5

Formulation as Finite-Volume-Method

numerical flux at

( )n 2/1in

2/1i

n

i

1n

i ggx

tuu +

+

=

n

2/1ig + 2/1ix +

tn

tn+1

xi-1/2 xi+1/2 xi+3/2

xi-1 xi xi+1x

t

6

Numerical flux of the Godunov-method

Properties

exact conservation

nonlinear wave propagation incorporated

adaptivity

))u,u;0(u(fg1iiRP

n

2/1i ++ =


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7

Approximate Riemann Solution

Definition: w=w(x/t;ul,ur) is defined to be an approximate

Riemann solution, if the following is valid:

1. al smallest, arlargest signal velocity

2. Consistency with the integral conservation

3. Consistency with the integral entropy condition

( ) ( ) ( ) ( )( )lrrl2x

2xrl ufuftuu

2

xdxu,u;txw +

=

( )( ) ( ) ( ) ( )( )lrrl

2x

2x rluFuFtUU

2

xdxu,u;txwU +

small enought

8

Approximate Riemann Solution

Definition: w=w(x/t;ul,ur) is defined to be an approximate

Riemann solution, if the following is valid:

1. al smallest, arlargest signal velocity

2. Consistency with the integral conservation

3. Consistency with the integral entropy condition

( ) ( ) ( ) ( )( )lrrl2x

2xrl ufuftuu

2

xdxu,u;txw +

=

( )( ) ( ) ( ) ( )( )lrrl2x

2xrl uFuFtUU

2

xdxu,u;txwU +

small enought

ur

u2

u1

ul

x2x

2x

{t

t


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9

Godunov-Type Methods

Definition: A method is calledGodunov-Type Scheme, if it

satisies

where w is an approximative Riemann solution.

Theorem: A Godunov-type scheme may be written in the

conservation form with the numerical flux:

( ) ( )dxu,u;txwx

1dxu,u;txw

x

1u

0

2x

n

1i

n

i

2x

0

n

i

n

1i

1n

i +

+

+

=

( ) ( ) i0

2x

n

1i

n

ii21i ut2

xdxu,u;txw

x

1ufg

+

= ++

10

The Roe Scheme

Roe-Linearization, P. Roe (1981):

1. Alr(u,u) = A(u)

2. Alrdiagonalizable

3. Mean value property:

( )

>

=otherwise

afora 1,33,11,3~ a

16

The Method of Harten, Lax and Van Leer

x

tl

at

x= ra

t

x=

2

x

2

x

lru

Simplest Godunov-type scheme

t


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17

HLL Approximate Riemann Solution

stateteintermediaa

velocityvelargest waandsmallesta,a

at

xfora

atxafora

at

xfora

u;u;txw

lr

rl

rr

rllr

ll

rl

>

ff

20

Riemann solver

Godunov-scheme Roe-scheme HLL-scheme(Harten, Lax, van Leer)

Exact solution of Exact solution of A priori estimates of

Riemann problem, lin. Riemann problem, the fastest wave speeds,

fixed point problem theory of characteristics

t

x

rarefaction

contact

discontinuity

shock wave

ur

u2u

1

ul

u2

t

xur

u1

ul

t

x

x = alt

ur

u1r

ul

x = arttax 1=

tax 2=

tax 3=


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8. Flux-Vector Splitting Methods

Splitting of the flux

with

non-negative EW

non-positive EW

Numerical flux

( ) ( ) ( )ufufuf + +=

left right

( ) ( )du

ufduA

++ =

( ) ( )du

ufduA

=

11, ++

+ += iiii ufufuug

22

Flux-Vector Splitting of Steger, WarmingEuler Theorem:

Flux of Euler equations is homogenious of degree 1:

Idea: Diagonalization of the Jacobian matrix A(u), splitting of the

diagonal matrices into a positive and a negative one

ufuf = allfor

uuAuf = ( )du

ufduA


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Diagonalization

Splitting of

Splitting of flux:

RRA = 1 ( )cvvcvdiagwith += ,,:

,+ +=

uuAuf

( )++++ = 321 ,,: aaadiagwith

( ) = 321 ,,: aaadiagwith

11, ++

+ += iiii ufufuug

24

( )( )

( )

( ) ( )( )1-2

caa

2

cvaa

2

vvff

vff

2

caavff

aaa122

f

2

141

14

2

2

2

11,11,4

21,11,3

1411,11,2

4121,1

++++

+

++

Steger-Warming flux into x-direction


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MnumberMach

( )

( )

( ) 2cv1-Cmit

vF2

1

12

CFF

vFF

CFF

c

vM,1M

4

cF

11

21,3211,21,4

21,11,3

11,11,2

12

1,1

+=

+

=

=

=

=+=

++

++

++

+ ( )

( )

( ) 2cv1-Cmit

vF2

1

12

CFF

vFF

CFF

c

vM,M1

4

cF

11

21,3211,21,4

21,11,3

11,11,2

12

1,1

=

+

=

=

=

==

1M

righttheFlux to

11 FF =+

lefttheFlux to

0F-1 =

1M1-

CFD03 Flux Functions

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