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Computational Fluid Dynamics


http://www.nd.edu/~gtryggva/CFD-Course/

Grétar Tryggvason

Lecture 7February 6, 2017

Computational Fluid Dynamicshttp://www.nd.edu/~gtryggva/CFD-Course/

Classical Methods

Discontinuous solutions

Entropy conditions

Shock speed

Conservative discretization


Classical Methodsfor Hyperbolic

Equations


�

∂ 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

Methods for Advection


Forward in Time, Centered in Space

(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+1n

n+1

Methods for AdvectionComputational Fluid Dynamics

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 jn

n+1 This scheme is O(Δt, h) accurate.

Another scheme for

Flow direction


To examine the stability we use the von Neuman’s 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

Another way: Find 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


fjn+1 = fj

n −ΔthU ( fj

n − fj−1n )

j-1 jn

n+1

O(Δt, h) accurate.

For the linear advection equation:

Flow direction

UΔth

≤ 1

Uh

The Upwind Scheme

�

∂f∂t

+U ∂f∂x

= 0



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


Finite Volume point of view:

xj-1/2 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



Stability in terms of Fluxes

for the upwind scheme


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

Stability in terms of fluxes


fi+1

Consider the following initial conditions:

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


Δ th

= 1.5 ⋅U = 1.5


f j+11

f j−1 f j

Fj−1 / 2 = Ufj−1n =U

Fj+1 / 2 = Ufjn = 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


Δ th

= 1.5 ⋅U = 1.5


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

Stability in terms of fluxesΔ 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).



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;

Stability in terms of fluxesComputational Fluid Dynamics

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

f jn+1 − 1

2 f j+1n + f j−1

n( )Δt

+U2h

f j+1n − f j−1

n( ) = 0


f jn= 12 f j+1

n + f j−1n( )

f jn+1 − f j

n

Δt+U2h

f j+1n − f j−1

n( ) = 0

Start with the unstable centered difference approximation

Replace the old value by

Resulting in


Lax-Fredrichs method- stable for λ < 1- 1st order in time and space- Conditionally consistent, but!

Uh2

1λ−λ

⎛

⎝⎜

⎞

⎠⎟ fxx +

Uh2

31−λ 2( ) fxxx

12h2

Δt−UΔt

⎛

⎝⎜

⎞

⎠⎟ fxx +

13Uh2 −U 3Δt2( ) fxxx

Error term:

λ =UΔth

hλ=

h2

UΔt


Δt ~ h

O Δt,h( )

If we assume that then we have

Used as a starting point for higher order methods


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

xf

Utf 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)

λ =UΔth



+Δ∂∂+Δ

∂∂+Δ

∂∂+=Δ+

62)()(

3

3

32

2

2 ttft

tft

tftfttf

Lax-Wendroff’s Method (LW-I)

First expand the solution in time

Then use the original equation to rewrite the time derivatives

xf

Utf

∂∂−=

∂∂

2

22

2

2

xf

Utf

xU

xf

Utt

ftt

f∂∂=

∂∂

∂∂−=⎟

⎠⎞⎜

⎝⎛

∂∂

∂∂−=⎟

⎠⎞⎜

⎝⎛∂∂

∂∂=

∂∂


)(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-Wendroff’s 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 j

n+1 =12

f 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:

f j

n+1 = f jn − λ f j

n − f j−1n( ) + 1

2λ(λ −1) f j

n − 2 f j−1n + f j−2

n( )- Stable if- Second-order accurate in time and space

20 ≤≤ λ


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



FTCSUnconditionally

Unstable

UpwindStable for

ImplicitUnconditionally

Stable

Lax-Friedrichs Conditionally consistent

Stable for

0=+ xt Uff

02

111

=−

+Δ− −+

+

hff

Utff n

jnj

nj

nj ( ) xxxxx f

Uhf

Ut 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 fUh

fUh 2

2

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

SummaryComputational Fluid Dynamics

Leap FrogStable for

Lax-Wendroff IStable for

Lax-Wendroff IISame as LW-I

Stable for

MacCormackSame 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


Beam-WarmingStable for

QUICKStable 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

�

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!

SummaryComputational Fluid Dynamics

Higher order upwindQUICK


s=1f1

s=2f2

s=3f3

s=4f4

s=5/2

At s = 5/2

�

∂f 2

∂x⎞

⎠ ⎟ i

≈ 1h

fi+1/ 22 − fi−1/ 2

2{ }

= 164h

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

f5 / 2 = 1/8( ) 3 f3 + 6 f2 − f1[ ] �

∂f∂t

+ 12∂f 2

∂x= D∂ 2 f

∂x 2

QUICK (Quadratic Upstream Interpolation for Convective Kinematics)

Use to solve:


Centered

UpwindQUICK

�

∂f∂t

+ 12∂f 2

∂x= D∂ 2 f

∂x 2


0 5 10 15 20 25 30 35 400

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 35 400

1

2

3

4

5

6

7

8

9

10

Re_cell=20Re_cell=10

�

∂f∂t

+ 12∂f 2

∂x= D∂ 2 f

∂x 2

Centered

UpwindQUICK


Discontinuoussolutions—shocks


�

∂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

xDiscontinuity of solution is allowed

( )RLR

L ffxxfxxf

xf >⎩⎨⎧

><

=0

0)0,(

�

dxdt

=U

Discontinuous SolutionsComputational Fluid Dynamics

( )RLR

L ffxxfxxf

xf >⎩⎨⎧

><

=0

0)0,(0=∂∂+

∂∂

xf

ftf

;0; ==dtdf

fdtdx

Inviscid Burgers’ Equation

t

x

Characteristics

0x

Lf

Rf

??=dtdx

0x

Shock Wave

Discontinuous Solutions


( ) ( ) axa

axfax

ffff

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


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( )


The Entropy Conditions


0=∂∂+

∂∂

xf

ftf

Inviscid Burgers’ Equation

The transformation

t

x0x

�

x→−x; t→−tLeaves the equation unchanged but results in an unphysical solution.

The entropy condition is used to select the correct solution

t

x0x

The Entropy ConditionsComputational Fluid Dynamics

( )RLR

L ffxxfxxf

xf <⎩⎨⎧

><

=0

0)0,(0=∂∂+

∂∂

xf

ftf

;0; ==dtdf

fdtdx

Reverse Shock (?)

Characteristics

0x

Lf

Rf t

x0x

Unstable, entropy-violatingsolution



( )RLR

L ffxxfxxf

xf <⎩⎨⎧

><

=0

0)0,(0=∂∂+

∂∂

xf

ftf

;0; ==dtdf

fdtdx

Rarefaction Wave (physically correct solution)

Characteristics

0x

Lf

Rf t

x0x

The Entropy ConditionsComputational Fluid Dynamics

Weak 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

xf

ftf

∂∂=

∂∂+

∂∂ ε

Therefore, what we are truly seeking is the solution to the viscous Burgers’ 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


The Entropy Condition states that the characteristics must “enter” the discontinuity. Thus, its speed C satisfies must satisfy

Given a conservation equation

)()( RL fFCfF ′>>′

0)( =∂

∂+∂∂

xfF

tf

Entropy Condition

t

x

Shock Wave

�

∂f∂t

+ ∂F∂f

∂f∂x

= 0

�

dxdt

= ∂F∂f

= ′ F ( f )

Rewrite in “characteristic” form

where:

�

dtds

=1; dxds

= ∂F∂f

or:


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 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

Conservative SchemesComputational Fluid Dynamics

�

∂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

Conservative Schemes


�

∂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 quantitiesConsider:

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!


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 discontinuousInitial data, conservative methods are essential (Lax).

Conservation and shock speed

Example: inviscid Burgers equation:

021

or0 2 =⎟⎠⎞⎜

⎝⎛

∂∂+

∂∂=

∂∂+

∂∂

fxt

fxf

ftf

Conservative MethodComputational Fluid Dynamics

Consider upwind and forward Euler scheme:

Example: for inviscid Burgers equationwith 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


Advecting a shock withseveral schemes


Example Problem: Linear Wave Equation

0,,0 ≥∞<<∞−=∂∂+

∂∂

txxf

Utf

⎩⎨⎧

><

=0001

)0,(xx

xf

Exact Solution: )( Utxf −

Apply various numerical methods


fjn+1 = fj

n −ΔthU ( fj

n − fj−1n )

�

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( )

f jn+1 = f j

n−1 −UΔth

f j+1n − f j−1

n( )

( )njnj

nj

tj ff

htUff −Δ−= +1

( )⎥⎦⎤

⎢⎣⎡ −Δ−+= −

+ tj

tj

tj

nj

nj ff

htUfff 1

1

21

Upwind

Leap-frog

Lax-Wendroff

MacCormack


Comparison: h=0.1; dt= 0.05;n=41, time=2.75

5 10 15 20 25 30 35 40-0.5

0

0.5

1

1.5

Leap-frog

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

Lax-Wendroff

5 10 15 20 25 30 35 40-0.5

0

0.5

1

1.5

MacCormack


Comparison: h=0.05; dt= 0.025;n=81, time=2.75

10 20 30 40 50 60 70 80-0.5

0

0.5

1

1.5

Lax-Wendroff

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

Leap-frog

10 20 30 40 50 60 70 80-0.5

0

0.5

1

1.5

MacCormack


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


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 Dynamicshttp://www.nd.edu/~gtryggva/CFD-Course/

Classical Methods

Discontinuous solutions

Entropy conditions

Shock speed


computational fluid - university of notre damegtryggva/cfd-course2017/lecture-7-2017.pdf ·...

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