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Computational Fluid Dynamics I

Numerical Methods!for Hyperbolic!EquationsIV!

Grtar Tryggvason !Spring 2009!

http://users.wpi.edu/~gretar/me612.html!Computational Fluid Dynamics I

Flows with shocks. Solutions to Burgers eqn. Entropy condition!

Conservation and shock speed!

Computational Fluid Dynamics I

Discontinuous!solutionsshocks!

Computational Fluid Dynamics I

ft

+U fx

= 0

The analytic solution is obtained by characteristics!

dxdt

=U; dfdt

= 0;

Consider the linear Advection Equation!

t

f

f

x

Discontinuity of solution is allowed!

( )RLR

L ffxxfxxf

xf >

>

>>

0)( =

+

xfF

tf

Entropy Condition!

t

x

Shock Wave!

ft

+ Ff

fx

= 0

dxdt

= Ff

= F ( f )

Rewrite in characteristic form!

where:!

dtds

=1; dxds

= Ff

or:!

Computational Fluid Dynamics I

Similarly, the shock speed is given by!

( ) ( ) ( ) ( )R

R

L

L

fffFfFC

fffFfF

Thus!

Entropy Condition!

t

x

Shock Wave!

C = FR FLfR fL

fL f fRfor!

Means that the hypothetical shock speed for values of f between the left and the right state must give shock speeds that are larger on the left and smaller on the right.!

Computational Fluid Dynamics I

Shock Speed!

Computational Fluid Dynamics I

x

ft

=ft

+fx'

x't

ft

+Fx

= 0

ft

C fx'

+Fx'

= 0

x' = x Ct

x't

= C

x

The speed of the shock! Write:!

Substitute into:!

where!

ft

C fx'

+Fx'

0 dx = 0

ft

0 dx C fx'

0 dx + Fx'

0 dx = 0

0

Computational Fluid Dynamics I

C fx'

0 dx + Fx'

0 dx = 0

C fR fL( ) + FR FL( ) = 0

C = FR FLfR fL

x

x

Rankine-Hugoniot relations!

Computational Fluid Dynamics I

ft

+ f fx

= 0

Example!

F = 12f 2

C = FR FLfR fL

=12fR2 fL2

fR fL=12

fR fL( ) fR + fL( )fR fL

C = 12fR + fL( )

Computational Fluid Dynamics I

Numerical Methods!for Hyperbolic!EquationsV!

http://users.wpi.edu/~gretar/me612.html!

Grtar Tryggvason !Spring 2009!

Computational Fluid Dynamics I

Flows with shocks. Solutions to Burgers eqn. Entropy condition!

Conservation and shock speed!

Advecting a shock with several schemes!

Godunovs Theorem!

Godunovs Method!

Higher Order Upwind schemes!

Artificial Viscosity!

Computational Fluid Dynamics I

Advecting a shock with!several schemes!

Computational Fluid Dynamics I

Example Problem: Linear Wave Equation!

0,,0

Computational Fluid Dynamics I

Comparison:! n=41 and n=81, time=2.75

10 20 30 40 50 60 70 80-0.5

0

0.5

1

1.5

Upwind

10 20 30 40 50 60 70 80-0.5

0

0.5

1

1.5

MacCormack

5 10 15 20 25 30 35 40-0.5

0

0.5

1

1.5

Upwind

5 10 15 20 25 30 35 40-0.5

0

0.5

1

1.5

MacCormack

Computational Fluid Dynamics I

First-Order Methods and Diffusion!

Upwind:!

( ) ( ) xxxxx fUhfUhxfU

tf 132

61

22

2

+=+

Lax-Wendroff:!

( ) ( ) xxxxxxx fUhfUhxfU

tf 2322 1

81

2 =

+

Dispersive = Wiggles!

Dissipative = Smearing!

= Uth

Computational Fluid Dynamics I

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).!

Computational Fluid Dynamics I

Godunov Theorem!

Computational Fluid Dynamics I

Question: How can we avoid oscillatory behavior?!(How can we preserve monotonicity)!

Godunov Theorem (1959):!

Monotone behavior of a numerical solution cannot be assured for linear finite-difference methods with more than first-order accuracy.!

Computational Fluid Dynamics I

Godunov Theorem - 1!

0=+

xfU

tf

For linear equation!

which yields:!2

22

2

2

;xfU

tf

xfU

tf

=

=

A general linear scheme can be written as:!

++ =k

nkjk

nj fcf1

Expanding in Taylor series around!n kjf +njf

)(2

32

222

hOxfhk

xfkhff nj

nkj +

++=+

Computational Fluid Dynamics I

Godunov Theorem - 2!

Substituting!

Also, can be Taylor expanded in time around!

++

+=+

k kk

kkk

nj

nj hOckx

fhkcxfhcff )(

232

2

221

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

2;;1

=== h

tUckhtUkcc

kk

kk

kk

1+njf

njf

)(2

32

221 tO

tft

tftff nj

nj +

++=+

)(2

32

222 tO

xftU

xftUf nj +

+=

(a)!

(b)!

Computational Fluid Dynamics I

Godunov Theorem - 3!

Condition for monotonicity:!11

11 then,If++

++ >>nj

nj

nj

nj ffff

Since! ++ =k

nkjk

nj fcf1

( ) ++++++ =k

nkj

nkjk

nj

nj ffcff 1

111

> 0 The above should be valid for arbitrary!

00 111 >>++

+nj

njk ffc

njf

Is it possible?!

Computational Fluid Dynamics I

Godunov Theorem - 4!

Hence we get:!

22;;1

=== h

tUckhtUkcc

kk

kk

kk

ckk

k2ckk

= kck

k

2

ek2

k

k2ek2

k

= kek

2

k

2

Define:! )0(2 >= kkk cec

This violates Cauchy inequality! a a( ) b b( ) a b( )2

Monotone 2nd Order scheme is impossible! !

a

b

Computational Fluid Dynamics I

The key word in Godunovs theorem is linear. To overcome its limitations, look for nonlinear scheames:!

1. Use central differencing and introduce numerical!!viscosity where needed (Artificial viscosity)!

2. Limit the fluxes in such a way that oscillations are avoided (high-order Godunov methods, flux-corrected transport, TVD shemes, etc.)!

Computational Fluid Dynamics I

Godunovs Method!

Computational Fluid Dynamics I

Godunov Method Finite Volume Method + Riemann solver!

ft

+x

F( f )[ ] = 0

Consider a 1-D Conservation Eq:!

Integrating across the j-th cell:!

Godunov Method!

Fj-1/2 Fj+1/2 !

1+jff j1

jf

x!j ! j+1 !j-1 !

0)()( 2/12/12/1

2/1=+

++

jj

x

xxFxFfdx

tj

j

Define the cell-average:! fav =1x j

f dxxj1/2

x j+1/2x j

favt

+ F(x j+1/2 ) F(x j1/2 ) = 0

x j = x j+1/2 x j1/2

Computational Fluid Dynamics I

Integrating over time!

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

favn+1 = fav

n tx j

1t

Fj+1/2 dt 1t

Fj1/2 dttt+t

tt+t

t

[ ] 0)( =+

fF

xtf

>

=++

+

2/11

2/1)0,(jj

jj

xxfxxf

xf 1+jfjf

x!j ! j+1 !

t!

2/1+= jcdtdx

t+t !

Godunov Method!Computational Fluid Dynamics I

Wave Diagram: !

1+jff j1

jf

j ! j+1 !j-1 !

Shock!Shock!Rarefaction!

Wave!

Godunov Method!

Computational Fluid Dynamics I

Godunov Method: The Procedure !

1. Construct a cell average value !

fav =1x j

f dxxj1/2

x j+1/22. Solve a Riemann problem to find the time ! integration of fluxes!

3. Construct new!

4. Go to 1. !

avf

ttt +=

Godunov Method!Computational Fluid Dynamics I

Special Case: Linear Advection Equation!

0,0 >=+

U

xfU

tf

for which the Riemann problem is trivial and we get!

( )njnjn javn jav ffhtUff 1,

1, + =

jj UfF =+ 2/1

1st Order Upwind Scheme!

Godunov Method!

Computational Fluid Dynamics I

fi+1

Consider the following initial conditions:!

1f j1 fi

Fj1 / 2 = Ufj1n = U

Fj+1 / 2 = Ufjn = 0

f jn+1 = fj

n th(Fj+1 / 2

n Fj1 / 2n )

Godunov Method!Computational Fluid Dynamics I

Godunov Method!

AdvectAverageAdvectAverageetc!

The averaging leads to numerical diffusion, even though the advection is exact!!

Computational Fluid Dynamics I

Godunov Method!

Numerical versus exact solution!

Computational Fluid Dynamics I

In its basic form upwind schemes are limited to relatively simple problems. !

By using Flux Vector Splitting or Gudonovs method, upwinding can be applied to nonlinear systems of hyperbolic equations.!

These methods are, however, only first order accurate in time and space.!

Godunov Method!

Computational Fluid Dynamics I

For nonlinear equations and systems the full Rieman problem must be solved!Since only the fluxes are needed, often approximate solvers are used!

Computational Fluid Dynamics I

Numerical Methods!for Hyperbolic!EquationsVI!

http://users.wpi.edu/~gretar/me612.html!

Grtar Tryggvason !Spring 2009!

Computational Fluid Dynamics I

Flows with shocks. Solutions to Burgers eqn. Entropy condition!

Conservation and shock speed!

Advecti