numerical methods for hyperbolic...

Computational Fluid Dynamics I

Numerical Methods!for Hyperbolic!Equations—I!

Grétar Tryggvason!Spring 2009!

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

FTCS and upwind! Stability in terms of fluxes!

Generalized upwind!Second order schemes for smooth flow!

Modified Equation!Conservation!


�

∂ 2 f∂t 2

− c 2 ∂2 f

∂x 2= 0

�

∂u∂t

− c 2 ∂v∂x

= 0

∂v∂t

− ∂u∂x

= 0

The wave equation:!

Write as:!

In general:!

∂u∂t∂v∂t

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

+a11 a12a21 a22

⎡

⎣⎢⎢

⎤

⎦⎥⎥

∂u∂x∂v∂x

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

= 0

Most of the issues involved can be addressed by examining:!

�

∂f∂t

+U ∂f∂x

= 0


FTCS and Upwind!


We will start by examining the linear advection equation:!∂f∂t

+ U∂f∂x

= 0

The characteristic for this equation are:!dxdt

=U; dfdt

= 0;

Showing that the initial conditions are simply advected by a constant velocity U!

t

f

f

x


A simple forward in time, centered in space discretization yields!

∂f∂t

+ U∂f∂x

= 0

fjn+1 = fj

n −Δt2hU ( fj+1

n − fj−1n )

j-1 j j+1 n

n+1


This scheme is O(Δt, h2) accurate, but a stability analysis shows that the error grows as!

ε n+1

ε n= 1− i

UΔt2h

sin kh

Since the amplification factor has the form 1+i() the absolute value of this complex number is always larger than unity and the method is unconditionally unstable for this case.!

iUΔt2h

sinkh

1

ε n+1

ε n


A simple forward in time but “upwind” in space discretization yields!

∂f∂t

+ U∂f∂x

= 0

fjn+1 = fj

n −ΔthU ( fj

n − fj−1n )

j-1 j n

n+1 This scheme is O(Δt, h) accurate. !

Another scheme for!

Flow direction!


To examine the stability we use the von Neumanʼs method:!

ε jn+1 − ε j

n

Δt+Uh(ε j

n − ε j−1n ) = 0

ε jn = εneikx j

ε n+1 − ε n

Δt+U ε n

h(1− e− ikh) = 0

The evolution of the error is governed by:!

Write the error as:!

�

εn+1

εn=1−UΔt

h(1− e−ikh )

�

G = εn+1

εn=1− λ(1− e−ikh ), λ = UΔt

h

Amplification factor!

�

G =1− λ + λe− ikhOr:!

�

G <1Need to find when!


Stable!

Im(G)

1

1

kh

λ 1-λ

G

UΔth

≤ 1

This restriction was first derived by Courant, Fredrik, and Levy in 1932, and is usually called the Courant condition, or the CFL condition. !

�

G =1− λ + λe− ikh, λ = UΔth

Stability condition: λ<1!

Graphically:!


�

G =1− λ + λe− ikh =1− λ + λcoskh − iλ sinkh

�

G 2 = 1− λ + λcoskh( )2 + λ2 sin2 kh

= 1− λ( )2 + 2 1− λ( )λcoskh + λ2 cos2 kh + λ2 sin2 kh

= 1− λ( )2 + 2 1− λ( )λcoskh + λ2

=1− 2λ + 2λ2 + 2 1− λ( )λcoskh

�

G 2 =1− 2λ 1− λ( ) if coskh = 0

G 2 =1 if coskh =1

G 2 =1− λ 4 − 3λ( ) if coskh = −1

�

G 2 ≤1 if λ ≤1

Finf the absolute value of the amplification factor!

1!

�

G 2

�

λ1!�

coskh = 0

�

coskh = −1


The CFL condition implies that a signal has to travel less than one grid spacing in one time step!

�

UΔt ≤ h

Flow direction!

j-1 j

n

n+1

Allowable characteristics!

Not allowable characteristics!


Stability in !terms of Fluxes!


f j−1 f j f j+1

Fj−1 / 2 = Ufj−1n = 1

Fj+1 / 2 = Ufjn = 0

Consider the following initial conditions:!

1

During one time step, U∆t of f flows into cell j, increasing the average value of f by U∆t/h.!

U = 1

Δ th

= 1.5 ⋅U = 1.5


If U∆t/h > 1, the average value of f in cell j will be larger than in cell j-1. In the next step, f will flow out of cell j in both directions, creating a larger negative value of f. Taking a third step will result in an even larger positive value, and so on until the compute encounters a NaN (Not a Number).!


fi+1


1f j−1 fi

Fj−1 / 2 = Ufj−1n = U

Fj+1 / 2 = Ufjn = 0

f jn+1 = fj

n −Δth(Fj+1 / 2

n − Fj−1 / 2n ) = 0 − 1.5(0 −1) = 1.5


f j+11

f j−1 f j

Fj−1 / 2 = Ufj−1n =U Fj+1 / 2 = Ufj

n = 1.5U

f jn+1 = fj

n −Δth(Fj+1 / 2

n − Fj−1 / 2n ) = 0 − 1.5(1.5 − 1) = 0.75

f j+1n+1 = fj+1

n −Δth(Fj+ 3/ 2

n − Fj+1/ 2n ) = 0 −1.5(0 −1.5) = 2.25


f j+11

f j−1 f j

Fj−1 / 2 = Ufj−1n =U

Fj+1 / 2 = Ufjn = 0.75U

Taking a third step will result in an even larger positive value, and so on until the compute encounters a NaN (Not a Number).!

Fj+1 / 2 = Ufjn = 2.25U


MOVIE FROM MATLAB!% one-dimensional advection by first order upwind.!n=80; nstep=100; dt=0.0125; length=2.0;!h=length/(n-1);y=zeros(n,1);f=zeros(n,1);f(1)=1.0;!

for m=1:nstep,m! hold off, plot(f); axis([1, n, -0.5, 1.5]);! pause(0.01);! y=f;! for i=2:n-1,! f(i)=y(i)-(dt/h)*(y(i)-y(i-1)); %upwind! end;!end;!


fj+1


1fj−1 fj

Fj−1/ 2 =U2( fj−1

n + fjn ) =1.0 Fj+1/ 2 =

U2( fj

n + f j+1n ) = 0.5

fjn+1 = fj

n −Δth(Fj+1/ 2

n − Fj−1/ 2n ) = 1.0 − 0.5(0.5 −1) =1.25

So cell j will overflow immediately!!

By considering the fluxes, it is easy to see why the centered difference approximation is always unstable.!


dt=0.25*h!

20 40 60 80-0.5

0

0.5

1

1.5

Upwind

Although the upwind method is exceptionally robust, its low accuracy in space and time makes it unsuitable for most serious computations !


Numerical Methods!for Hyperbolic!Equations—II!

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



FTCS and upwind! Stability in terms of fluxes!

Generalized upwind!Second order schemes for smooth flow!

Modified Equation!Conservation!


fjn+1 = fj

n −ΔthU ( fj

n − fj−1n )

j-1 j n

n+1

O(Δt, h) accurate. !

For the linear advection equation:!

Flow direction!

UΔth

≤ 1

U h

First Order Schemes!

The Upwind Scheme!

�

∂f∂t

+U ∂f∂x

= 0


Finite Volume point of view:!

xj1/2 xj+1/2 !

�

f j−1

�

f j

x!

f

�

f j+1

�

f jn+1 = f j

n − Δth(Fj+1/ 2

n − Fj−1/ 2n ) = f j

n − ΔthU( f j

n − f j−1n )

�

Fj+1/ 2=Uf jn

�

Fj−1/ 2=Uf j−1n

First Order Schemes!Computational Fluid Dynamics I

Generalized Upwind Scheme (for both U > 0 and U < 0 )!

�

f jn+1 = f j

n −UΔth( f j

n − f j−1n ), U > 0

�

f jn+1 = f j

n −UΔth( f j+1

n − f jn ), U < 0

Define:!

�

U + = 12U + U( ), U− = 1

2U − U( )

The two cases can be combined into a single expression:!

�

f jn+1 = f j

n − ΔthU +( f j

n − f j−1n ) +U−( f j+1

n − f jn )[ ]


Or, substituting! −+ UU ,

�

f jn+1 = f j

n −U Δt2h( f j+1

n − f j−1n ) +

U Δt2h

( f j+1n − 2 f j

n + f j−1n )

central difference + numerical viscosity! Dnum =U h2


Other First Order Schemes!


Implicit (Backward Euler) Method!

- Unconditionally stable!- 1st order in time, 2nd order in space!- Forms a tri-diagonal matrix (Thomas algorithm)!

( ) 02

11

11

1

=−+Δ− +

−++

+nj

nj

nj

nj ff

hU

tff

nj

nj

nj

nj f

tf

hUf

tf

hU

Δ=−

Δ+ +

−++

+

12

12

11

111

jnjj

njj

njj Cfbfdfa =++ +

−++

+11

111



Lax-Fredrichs method!

- stable for λ < 1!- 1st order in time, 1nd order in space!- Conditionally consistent!�

f jn+1 − 1

2 f j+1n + f j−1

n( ) f jnΔt

+ U2h

f j+1n − f j−1

n( ) = 0

( ) xxxxx fUhfUh 22

13

12

λλλ

−+⎟⎠⎞⎜

⎝⎛ −

Error term:!


λ =UΔth

hλ=

h2

UΔt


Second Order Schemes!


)(2

211

tOtff

tf n

jnj Δ+

Δ−

=∂∂ −+

Leap Frog Method!

The simplest stable second-order accurate (in time) method:!

Modified equation!

( )nj

nj

nj

nj ff

htUff 11

11−+

−+ −Δ−=

( ) +−=∂∂+

∂∂

xxxfUh

xfU

tf 1

62

2

λ

- Stable for ! 1<λ- Dispersive (no dissipation) – error will not damp out!- Initial conditions at two time levels !- Oscillatory solution in time (alternating) !

Second Order Schemes!Computational Fluid Dynamics I

+Δ∂∂+Δ

∂∂+Δ

∂∂+=Δ+

62)()(

3

3

32

2

2 ttft

tft

tftfttf

Lax-Wendroffʼs Method (LW-I)!

First expand the solution in time!

Then use the original equation to rewrite the time derivatives!

xfU

tf

∂∂−=

∂∂

2

22

2

2

xfU

tf

xU

xfU

ttf

ttf

∂∂=

∂∂

∂∂−=⎟

⎠⎞⎜

⎝⎛

∂∂

∂∂−=⎟

⎠⎞⎜

⎝⎛∂∂

∂∂=

∂∂



)(2

)()( 32

2

22 tOtxfUt

xfUtfttf Δ+Δ

∂∂+Δ

∂∂−=Δ+

Substituting!

Using central differences for the spatial derivatives!

�

f jn+1 = f j

n −UΔt2h

f j+1n − f j−1

n( ) + U2Δt 2

2h2f j+1n − 2 f j

n + f j−1n( )

2nd order accurate in space and time!

Stable for! 1<ΔhtU


Two-Step Lax-Wendroffʼs Method (LW-II)!

LW-I into two steps:!

For the linear equations, LW-II is identical to LW-I!

02/

2/)( 112/12/1 =

−+

Δ+− ++

++

hff

Ut

fff nj

nj

nj

nj

nj

02/12/1

2/12/1

1

=−

+Δ− +

−++

+

hff

Utff n

jnj

nj

nj

Step 1 (Lax)!

Step 2 (Leapfrog)!

- Stable for ! 1/ <Δ htU- Second order accurate in time and space !



MacCormack Method!

Similar to LW-II, without !

( )njnj

nj

tj ff

htUff −Δ−= +1

( )⎥⎦⎤

⎢⎣⎡ −Δ−+= −

+ tj

tj

tj

nj

nj ff

htUfff 1

1

21

Predictor!

Corrector!

- A fractional step method!!- Predictor: forward differencing!!- Corrector: backward differencing!

- For linear problems, accuracy and stability ! properties are identical to LW-I. !

2/1,2/1 −+ jj


Second-Order Upwind Method!

Warming and Beam (1975) – Upwind for both steps!

( )nj

nj

nj

tj ff

htUff 1−−Δ−=

�

f jn+1 = 1

2f jn + f j

t −U Δth

f jt − f j−1

t( ) −U Δth

f jn − 2 f j−1

n + f j−2n( )⎡

⎣ ⎢ ⎤ ⎦ ⎥

Predictor! Corrector!

Combining the two:!

( ) ( )nj

nj

nj

nj

nj

nj

nj fffffff 2111 2)1(

21

−−−+ +−−+−−= λλλ

- Stable if!- Second-order accurate in time and space !

20 ≤≤ λ



QUICK!s=1!f1!

s=2!f2!

s=3!f3!

s=4!f4!

s=5/2!

�

1− 2 : (2 − s) f1 + (s−1) f22 − 3 : (3− s) f2 + (s− 2) f3

�

1− 3 : 12(3− s) (2 − s) f1 + (s−1) f2[ ] + 1

2(s−1) (3− s) f2 + (s− 2) f3[ ]

�

s = 5 /2 : f5 / 2 = 18(3 f3 + 6 f2 − f1)

Interpolate linearly between 1-2 and 2-3!

Then quadratically between 1-2!

The function can now be evaluated at the half point!

Third Order Schemes!

�

∂f i∂x

= 1h

fi+1/ 2 − f i−1/ 2( )


QUICK!Approximate the derivative:!

�

∂f i∂x

= 1h

fi+1/ 2 − f i−1/ 2( )

where the value at the half points by second order upwind interpolation!

�

∂f∂x

⎞ ⎠ ⎟ i

≈ 18h

3 f i + 6 fi−1 − f i−2[ ] − 3 fi+1 + 6 f i − fi−1[ ]{ }

Third Order Schemes!


The one-step Lax-Wendroff is not easily extended to non-linear or multi-dimensional problems. The split version is.!

In the Lax-Wendroff and the MacCormack methods the spatial and the temporal discretization are not independent.!

Other methods have been developed where the time integration is independent of the spatial discretization, such as the Beam-Warming and various Runge-Kutta methods!


FTCS!Unconditionally

Unstable!

Upwind!Stable for!

Implicit!Unconditionally!

Stable!

Lax-Friedrichs! Conditionally consistent!

Stable for !

0=+ xt Uff

02

111

=−

+Δ− −+

+

hff

Utff n

jnj

nj

nj ( ) xxxxx fUhfUt 2

22

2162

λ+−Δ−

011

=−

+Δ− −

+

hff

Utff n

jnj

nj

nj

( )

( ) xxx

xx

fUh

fUh

1326

12

22

+−−

−

λλ

λ

1≤λ

( ) xxxxx fUhfUh 22

13

12

λλλ

−+⎟⎠⎞⎜

⎝⎛ −

( )0

2

11

11

1

=−

+Δ− +

−++

+

hff

Utff n

jnj

nj

nj

xxxxx ftUUhftU⎥⎦⎤

⎢⎣⎡ Δ+−Δ 232

2

31

61

2

( )

( )0

2

2/

11

111

=−

+

Δ+−

−+

−++

hff

U

tfff

nj

nj

nj

nj

nj

1≤λ

Summary!


Leap Frog!Stable for!

Lax-Wendroff I!Stable for!

Lax-Wendroff II!Same as LW-I!

Stable for!

MacCormack!Same as LW-I!

Stable for!

0=+ xt Uff

022

1111

=−

+Δ− −+

−+

hff

Utff n

jnj

nj

nj ( ) xxxf

Uh 16

22

−λ

( )

( ) xxxx

xxx

fUh

fUh

23

22

18

16

λλ

λ

−−

−−

1≤λ

1≤λ

1≤λ

( )

( )0

222

21122

111

=+−

Δ−

−+

Δ−

−+

−++

hfff

tU

hff

Utff

nj

nj

nj

nj

nj

nj

nj

02/

2/)( 112/12/1 =

−+

Δ+− ++

++

hff

Ut

fff nj

nj

nj

nj

nj

02/12/1

2/12/1

1

=−

+Δ− +

−++

+

hff

Utff n

jnj

nj

nj 1≤λ

( )01 =

−+

Δ− +

hff

Utff n

jnj

nj

tj

( ) ( )0

2/ 11

=−

+Δ+− −

+

hff

Utfff t

jtj

tj

nj

nj

Summary!Computational Fluid Dynamics I

Beam-Warming!Stable for!

QUICK!Stable for!

ENO!

WENO!

0=+ xt Uff

( )

( ) 022

243

212

2

211

=+−Δ−

+−+

Δ−

−−

−−+

nj

nj

nj

nj

nj

nj

nj

nj

fffhtU

hfff

Utff

20 ≤≤ λ

( )( )

( ) ( ) xxxx

xxx

fUh

fUh

λλ

λλ

−−−

−−

218

216

23

2

Summary!

�

f jn+1 − f j

n

Δt+

U3 f j

n + 6 f j−1n − f j−2

n( ) − 3 f j+1n + 6 f j

n − f j−1n( )

8h1≤λ

A large number of (conditionally) stable and accurate methods exists for hyperbolic equations with smooth solutions !

And Many More!!


Modified Equation!


Looking at the structure of the error terms—by deriving the Modified Equation—can often lead to insight into how the approximate solution behaves!


Derive modified equation for upwind difference method:!

0)( 1

1

=−+Δ−

−

+nj

nj

nj

nj ff

hU

tff

Using Taylor expansion:!

+Δ∂∂+Δ

∂∂+Δ

∂∂+=+

62

3

3

32

2

21 t

tft

tft

tfff n

jnj

+∂∂−

∂∂+

∂∂−=− 62

3

3

32

2

2

1

hxfh

xfh

xfff n

jnj


Substituting!

⎭⎬⎫

⎩⎨⎧

−⎥⎦⎤

⎢⎣⎡ +Δ

∂∂+Δ

∂∂+Δ

∂∂+

Δnj

nj ft

tft

tft

tff

t

621 3

3

32

2

2

062

3

3

32

2

2

=⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ +

∂∂−

∂∂+

∂∂−−+

hxfh

xfh

xfff

hU n

jnj

Therefore,!

+−Δ−+Δ−=∂∂+

∂∂

xxxtttxxtt fUhftfUhftxfU

tf

6622

22

It helps the interpretation if all terms are written in ! xxxxx ff ,


Taking further derivatives (first in time, then in space): !

+−Δ−+Δ−=+ xxxtttttxxttttxttt fUhftfUhftUff6622

22

++Δ+−Δ=−− xxxxtttxxxxttxxxtx fhUftUfhUftUfUUf6622

22222+!

⎟⎠⎞⎜

⎝⎛ +−Δ+

⎟⎠⎞⎜

⎝⎛ Δ++−Δ+=

)(22

)(22

2

2

hOuUuUx

tOfUftfUf

xxxxxt

ttxttt

xxtt


Similarly, we get !),(3 htOfUf xxxttt Δ+−=

),(2 htOfUf xxxttx Δ+=

),( htOUff xxxxxt Δ+−=

Final form of the modified equation: !

( ) ( ) xxxxx fUhfUhxfU

tf 132

61

22

2

+−−−=∂∂+

∂∂ λλλ

+O h3,h2Δt,hΔt 2 ,Δt 3⎡⎣ ⎤⎦

htUΔ=λ


Numerical dissipation (diffusion)!

( ) ( ) ++−−−=∂∂+

∂∂

xxxxx fUhfUhxfU

tf 132

61

22

2

λλλ

By applying upwind differencing, we are effectively solving: !

Also note that the CFL condition ! 1<Δ=htUλ

ensures a positive diffusion coefficient !

Dissipation!( ) xxf

Dispersion!( ) xxxf


�

∂f∂t

= α ∂ n f∂xn

�

f (x, t) = a(t)eikx

�

da(t)dt

= (ik)nαa(t)

�

a(t) = aoe( ik )nαt

For even n we get diffusion, for odd n we get dispersion!

Consider:!

Look for solutions of the form:!

Substitute to get!

solve!


First-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!

htUΔ=λ


Second-Order Methods and Dispersion!

Lax-Wendroff:!∂f∂t

+U ∂f∂x

= −Uh2

21− λ2( ) fxxx − Uh

3

8λ 1− λ2( ) fxxxx

Beam-Warming:!

∂f∂t

+U ∂f∂x

=Uh2

61− λ( ) 2 − λ( ) fxxx −

Uh3

81− λ( )2 2 − λ( ) fxxxx

Dispersive = Wiggles!

htUΔ=λ


dt=0.25*h!

20 40 60 80-0.5

0

0.5

1

1.5Lax-WendroffUpwind


Conservative discretization

schemes!


In finite volume method, equations in conservative forms are needed in order to satisfy conservation properties.!

As an example, consider a 1-D equation!

�

∂f (x, t)∂t

+∂F[ f (x, t)]

∂x= 0

where F denotes a general advection/diffusion term, e.g.!

�

F =12f 2; F = µ(x) ∂f

∂x


�

∂F∂x

dxL∫ =

Fj+1/ 2 − Fj−1/ 2

Δx∑ Δx

Xj-1/2 xj+1/2 !

�

f j+1

�

f j−1

�

f j

x!

In discretized form:!

�

= + Fj−1/ 2 − Fj−3 / 2 + Fj+1/ 2 − Fj−1/ 2 + Fj+3 / 2 − Fj+1/ 2 +[ ]

�

= FL − F0

If F = 0 at the end points of the domain, f is conserved.!

Integrating over the domain L,!

�

∂f∂tdx

L∫ +

∂F∂x

dxL∫ = 0

�

F(L) − F (0) = 0

�

⇒ddt

fdxL∫ = 0


�

∂f∂t

+ ∂∂x

12f 2

⎛ ⎝ ⎜

⎞ ⎠ ⎟ = 0

Examples of Conservative Form!

Discretize using upwind !

�

∂∂x

12f 2

⎛ ⎝ ⎜

⎞ ⎠ ⎟ ≈

12h

fi2 − f i−1

2( ) Conservative!�

∂f∂t

+ f ∂f∂x

= 0

�

f ∂f∂x

≈ f ih

f i − f i−1( ) Non-conservative!

versus!


�

∂F∂x

dxL∫ ≈ Fj+1/ 2 − Fj−1/ 2∑ =

+ fi−12 − f i−2

2 + f i2 − f i−1

2 + f i+12 − fi

2 +

�

∂F∂x

dxL∫ ≈ Fj+1/ 2 − Fj−1/ 2∑ =

+ fi−12 − f i−1 f i + f i

2 − f i f i−1 + f i+12 − f i+1 f i +

�

∂∂x

12f 2

⎛ ⎝ ⎜

⎞ ⎠ ⎟ ⎞ ⎠ ⎟ i

≈ 12h

fi2 − f i−1

2( )

�

f ∂f∂x

⎞ ⎠ ⎟ i

≈ f ih

fi − f i−1( )


�

∂f∂t

+ ∂∂x

µ(x) ∂f∂x

⎛ ⎝ ⎜

⎞ ⎠ ⎟ = 0

Another example (variable diffusion coefficient)!

Discretize !

�

∂∂x

µ(x) ∂f∂x

⎛ ⎝

⎞ ⎠

Conservative!

�

µ∂2 f∂x 2

+∂µ∂x

∂f∂x

Non-conservative!

versus!


�

∂f∂t

+ f ∂f∂x

= 0

�

f ∂f∂t

+ f ∂f∂x

⎛ ⎝ ⎜

⎞ ⎠ ⎟ = 0

�

∂∂t

f 2

2+ ∂∂x

f 3

3= 0

�

ddt

f 2

2dx∫ = − ∂

∂xf 3

3dx∫ = f 3

3⎡

⎣ ⎢

⎤

⎦ ⎥ −∞

∞

= 0

�

f ∞( ) = f −∞( ) = 0

�

ddt

f n

ndx∫ = 0

Similarly, it can be shown that!

Conservation of higher order quantities!Consider:!

Integrate:!


Generally it is believed that conservative schemes are superior to non-conservative ones and that the more conserved quantities the discrete approximation preserves, the better the scheme. While mostly true, there are exceptions as we will discuss later!!


Numerical Methods!for Hyperbolic!Equations—III!

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



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

Conservation and shock speed!

Advecting a shock with several schemes!

Godunovʼs Theorem!

Artificial Viscosity!

Multidimensional flows!


Discontinuous!Solutions!Shocks-I!


�

∂f∂t

+U ∂f∂x

= 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!

(Weak Solution)!

( )RLR

L ffxxfxxf

xf >⎩⎨⎧

><

=0

0)0,(

�

dxdt

=U


( )RLR

L ffxxfxxf

xf >⎩⎨⎧

><

=0

0)0,(0=∂∂+

∂∂

xff

tf

;0; ==dtdff

dtdx

Inviscid Burgersʼ 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

x 0x

Unstable, entropy-violating!solution!


( )RLR

L ffxxfxxf

xf <⎩⎨⎧

><

=0

0)0,(0=∂∂+

∂∂

xff

tf

;0; ==dtdff

dtdx

Rarefaction Wave (physically correct solution)!

Characteristics!

0x

Lf

Rf t

x 0x


Shock Speed!


x

∂f∂t

=∂f∂t

+∂f∂x'

∂x'∂t

∂f∂t

+∂F∂x

= 0

∂f∂t

− C ∂f∂x'

+∂F∂x'

= 0

x' = x − Ct

∂x'∂t

= C

x’ The speed of the shock! Write:!

Substitute into:!

where!

∂f∂t

− C ∂f∂x'

+∂F∂x'

⎛ ⎝

⎞ ⎠

Δ→ 0∫ dx = 0

�

∂f∂t

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Δ→0∫ dx − C ∂f

∂x'⎛ ⎝ ⎜

⎞ ⎠ ⎟

Δ→0∫ dx + ∂F

∂x'⎛ ⎝ ⎜

⎞ ⎠ ⎟

Δ→0∫ dx = 0

0


�

− C ∂f∂x'

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Δ→0∫ dx + ∂F

∂x'⎛ ⎝ ⎜

⎞ ⎠ ⎟

Δ→0∫ dx = 0

−C fR − fL( ) + FR − FL( ) = 0

C =FR − FLfR − fL

x

x’

Rankine-Hugoniot relations!


∂f∂t

+ f ∂f∂x

= 0

Example!

F =12f 2


=12fR2 − fL2

fR − fL=12

fR − fL( ) fR + fL( )fR − fL

C =12fR + fL( )


Discontinuous!Solutions!Shocks-II!


Weak solutions to hyperbolic equations may not be unique.!

How can we find a physical solution out of many weak!solutions?!

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

2

2

xf

xff

tf

∂∂=

∂∂+

∂∂ ε

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


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

−−≥≥

−−

0)( =∂

∂+∂∂

xfF

tf

Version I:!

Version II:!

And some others…!

Entropy Condition!

t

x

Shock Wave!

for!

fL ≥ f ≥ fR


Means that the characteristics on both side must “slope into” the shock!

Start with:!

)()( RL fFCfF ′>>′

0)( =∂

∂+∂∂

xfF

tf

The condition:!

Entropy Condition!

t

x

Shock Wave!

F ' = ∂F∂f

∂f∂t

+ F ' ∂F∂x

= 0where!

F 'RF 'L


Similarly, the shock speed is given by!

( ) ( ) ( ) ( )R

R

L

L

fffFfFC

fffFfF

−−≥≥

−−

Thus!

Entropy Condition!

t

x

Shock Wave!


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.!


In CFD, however, we do not want to solve the viscous!Burgersʼ equation with extremely small because it !becomes computationally expensive.!

Nevertheless, in many gas dynamics application, there!is need to capture discontinuous solution behavior.!

And we want to reproduce the weak solution behavior!numerically by solving inviscid Burgersʼ equation.!

ε

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


Examples: !!Godunov method!!Approximate Riemann solver!!Artificial dissipation!

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

Entropy Condition!


Conservative Schemes are guarantied to give the correct Shock Speed since they correspond to a direct application of the conservation principles. !

Non-conservative schemes may or may not do so.!


In capturing the correct solution behavior for discontinuous!Initial data, conservative methods are essential (Lax).!

Conservation and shock speed!

Example: inviscid Burgers equation:!

021or0 2 =⎟

⎠⎞⎜

⎝⎛

∂∂+

∂∂=

∂∂+

∂∂ f

xtf

xff

tf

Conservative Method!


Consider upwind and forward Euler scheme:!

Example: for inviscid Burgers equation!with discontinuous initial data!

0=+ xt fff

1−j

1

j 1+j

�

f jn+1 = f j

n − Δthf jn f j

n − f j−1n( ) = 0

Non-conservative form!

Never moves!!

�

ft + 12 f

2( )x = 0

�

f jn+1 = f j

n − Δt2h

( f jn )2 − ( f j−1

n )2( ) = Δt2h

Conservative form!

Conservative Method!

numerical methods for hyperbolic...

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