WPPII Computational Fluid Dynamics I - Unicampphoenics/SITE_PHOENICS/Apostilas/CFD-1_U... · WPPII Computational Fluid Dynamics I Numerical Methods for Wave Equations: Part I: Smooth

Computational Fluid Dynamics IPPPPIIIIWWWW

Numerical Methods forWave Equations:

Part I: Smooth 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


Method of Characteristics


0=∂∂+

∂∂

xfc

tf

The characteristics for this equation are:

;0; ==dtdfc

dtdx

1st Order Wave Equation

t

f

f

x

Computational Fluid Dynamics IPPPPIIIIWWWW1-D Wave Equation (2nd Order Hyperbolic PDE)

which leads to

02

22

2

2

=∂∂−

∂∂

xfc

tf

Define;;

xfw

tfv

∂∂=

∂∂=

0

02

=∂∂−

∂∂

=∂∂−

∂∂

xv

tw

xwc

tv

Computational Fluid Dynamics IPPPPIIIIWWWWIn matrix form

Find the eigenvalue, eigenvector

0or001

0 2

=+=

−

−+

xt

x

x

t

t

wvc

wv

Auu

Can it be transformed into the form

0or00

0=+=

+

xt

x

x

t

t

wv

wv

uu λλ

λ?

01

2

12

=

−−−−

ll

c λλ

( ) 0T =− qIA λ

Computational Fluid Dynamics IPPPPIIIIWWWWEigenvalue

cdtdxc ±===−=− λλλ ;0;0 22T IA

Eigenvector

−

==−−c

lcl1

;0 121 qFor ,c+=λ

==−

clcl

1;0 221 qFor ,c−=λ

The solution (v,w) is governed by ODE’salong the characteristic lines cdtdx ±=/

Computational Fluid Dynamics IPPPPIIIIWWWWOn ,/ cdtdx +=

( ) ( ) ( )xtxtxt vwcwcv −−−=+⋅ 2T1 Auuq

( ) ( ) 0=−=+−+=dtdwc

dtdvcwwccvv xtxt

If cwvr −≡1 (Riemann Invariant), we have

cdtdxdt

cwvd +==− on0)((If c = const)

Computational Fluid Dynamics IPPPPIIIIWWWWOn ,/ cdtdx −=

( ) ( ) ( )xtxtxt vwcwcv −+−=+⋅ 2T2 Auuq

( ) ( ) 0=+=−+−=dtdwc

dtdvcwwccvv xtxt

If cwvr +≡1 (Riemann Invariant), we have

cdtdxdt

cwvd −==+ on0)((If c = const)


Pcdtdx +=/

cdtdx −=/

cdtdxdtdwc

dtdv +==− on0

cdtdxdtdwc

dtdv −==+ on0

1

2

3

known,,xf

tff

∂∂

∂∂

dxfdtfdf xt +=


[ ][ ] [ ][ ])1()()1()(21)1()()1()(

21)1()( xPxwPwtPtvPvfPf −++−+=−

[ ][ ]

[ ][ ]

−+=−

−+=−

)1()()1()(21)1()(

)1()()1()(21)1()(

wPwcPcvPv

tPtcPcxPx

Finite difference approximation to determine )(Pf[ ]xt fwfv == ,

P cdtdx −=/

1

2cdtdx +=/

cdtdx +=on

[ ][ ]

[ ][ ]

−+−=−

−+−=−

)2()()2()(21)2()(

)2()()2()(21)2()(

wPwcPcvPv

tPtcPcxPxc

dtdx −=on

Nonlinear system – iterative procedure


Numerical Methods:Finite Volume Approach


When using finite volume approximations, we work directly with the integral form of the conservation principles. The average values of f over a small volume are stored

xj−1/2 xj+1/2

1−jfjf

x

f

1+jf

Computational Fluid Dynamics IPPPPIIIIWWWWIn finite volume method, equations in conservative forms are needed in order to satisfy conservation properties.

As an example, consider a 1-D equation

0)],([),( =∂

∂+∂

∂x

txfFt

txf

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

xfxFfF∂∂== )(;

21 2 µ


0=∂∂+

∂∂

∫∫LL

dxxFdx

tf

0)0()( =− FLF

0=⇒ ∫L

fdxdtd

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

Integrating over the domain L,


∑∫ ∆−

=∂∂ −+

xFF

dxxF jj

L

2/12/1

xj−1/2 xj+1/2

1+jf

1−jfjf

x

In discretized form:

[ ]!! +−+−+−+∆

= ++−+−− 2/12/32/12/12/32/11

jjjjjj FFFFFFx

[ ]01 FFx L −∆

=


0)(,021 2 =

∂∂

∂∂+

∂∂=

∂∂+

∂∂

xfx

xtff

xtf µ

Examples of Conservative Form

Discretize

( )( ) ( )( )[ ]11112

2/12/1

21

1

−−++

−+

−+−−+

=

∂∂−

∂∂

jjjjjjjj

jj

ffffh

xf

xf

h

µµµµ

µµ

∂∂

∂∂

xfx

x)(µ

Conservative

Computational Fluid Dynamics IPPPPIIIIWWWWExamples of Non-conservative Form

Discretize

0,0 2

2

=∂∂

∂∂+

∂∂+

∂∂=

∂∂+

∂∂

xf

xxf

tf

xff

tf µµ

xf

xxf

∂∂

∂∂+

∂∂ µµ 2

2

( ) ( )( )

( ) ( )

+−−+−+

=

−−+−+

−−+−−+++−+

−+−+−+

11111111112

1111112

4121

4121

jjjjjjjjjjjjjj

jjjjjjjj

fffffffh

fffffh

µµµµµµµ

µµµ

Non-conservative


0=∂∂+

∂∂

xF

tf

F = Uf Advection

Diffusion

Advection/Diffusion

xfDF∂∂=

xfDUfF∂∂−=

Finite Volume Method for Conservative Equations


0=∂∂+

∂∂

xF

tf

0=∂∂+

∂∂

∫∫∆∆

dxxFdx

tf

xx

02/12/1 =−+ −+∆∫ jjx

FFdxfdtd

0)( 2/12/1 =−+ −+ jjj FFhfdtd

Fj−1/2 Fj+1/2

1−jf

x

Finite Volume Formulation

j j+1j−1

jf1+jf


∂∂−

∂∂+−−=

−+−+

2/12/12/12/1 )(

jjjjj x

fxfDffUf

dtdh

)(21

12/1nj

njj fff +≈ ++

hff

xf n

jnj

j

−≈

∂∂ +

+

1

2/1

)(1 1 nj

njj ff

tf

dtd −

∆≈ +

Approximating

1-D Advection-Diffusion Equation

0=

∂∂−

∂∂+

∂∂

xfDUf

xtf

FVM Equation:


=−∆

+ )( 1 nj

nj ff

th

( )nj

nj

nj

nj

nj

nj

nj fff

hDff

hU

tff

11211

1

2)(2 −+−+

+

+−=−+∆−

Rearranging the terms:

Substituting:

Which is exactly the same as the finite difference equation if we take the average value to be the same as the value in the center of the cell

[ ]

−−

−++−+− −+

−+ hff

hff

DffffUnj

nj

nj

njn

jnj

nj

nj

1111 )(2

1)(21


Numerical Methods for 1-D Advection Equation:Stability Consideration

(Finite Difference Approach)


We will start by examining the linear advection equation:

0=∂∂+

∂∂

xfU

tf

The characteristic for this equation are:

;0; ==dtdfU

dtdx

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

t

f

f

x


A forward in time, centered in space (FTCS) discretization yields

0=∂∂+

∂∂

xfU

tf

)(2 11

1 nj

nj

nj

nj ffU

htff −+

+ −∆−=

j−1 j j+1n

n+1

Finite difference equation

Computational Fluid Dynamics IPPPPIIIIWWWWThis scheme is O(∆t, ∆x2) accurate, but a stability analysis shows that the error grows as

ε n+1

ε n =1− iU∆t2h

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.

iU∆t2h

sin kh

1

ε n+1

ε n


A forward in time but “upwind” (windward) in space discretization yields

)( 11 n

jnj

nj

nj ffU

htff −

+ −∆−=

j−1 jn

n+1 This scheme is O(∆t, ∆x) accurate.

0=∂∂+

∂∂

xfU

tf

Alternative Scheme: Upwind Difference1−jf jf 1+jf

U

Computational Fluid Dynamics IPPPPIIIIWWWWTo examine the stability we use the von Neuman’s method:

nj

nj

nj ff ε+=

0)( 1

1

=−+∆−

−

+nj

nj

nj

nj

hU

tεε

εε

jikxnnj eεε =

0)1(1

=−+∆− −

+ikh

nnn

eh

Ut

εεε

Substituting into the modified equation,


0)1(1

=−+∆− −

+ikh

nnn

eh

Ut

εεε

khih

tUkhh

tUeh

tU ikhn

n

sin)cos1(1)1(11 ∆−−∆−=−∆−= −+

εε

Amplification factor

khih

tUkhh

tUn

n

sin2

sin21 21 ∆−∆−=+

εεor


htUeG ikh ∆=−−= − λλ ),1(1

Amplification Factor

Stable

Re(G)

Im(G)

1

1

kh

λ1−λ

Stability Condition:

U∆th

≤ 1

CFL Condition(Courant-Friedrichs-Lewy;

1932)

G

1<λ

Computational Fluid Dynamics IPPPPIIIIWWWWImplication of the CFL Condition (Hirsch, vol.1, p. 288)

Stability

The domain of dependence of the differential equation should be contained in the domain of dependence of the discretized equations.


Stability Consideration(Finite Volume Approach)


0=∂∂+

∂∂

xF

tf

0=∂∂+

∂∂

∫∫∆∆

dxxFdx

tf

xx

02/12/1 =−+ −+∆∫ jjx

FFdxfdtd

( ) ( ) 02/12/11 =−∆+− −++ n

jnj

nj

nj FFtffh

Finite Volume Formulation

Fj-1/2 Fj+1/2

1+jf

f j−1

jf

xj j+1j-1

( )nj

nj

nj

nj FF

htff 2/12/1

1−+

+ −∆−=


UffF jjj )(21

12/1 +≈ ++

UfF jj ≈+ 2/1

Approximating the advective fluxes:

Taking the average (Central):

Upwind:

1+jf

f j−1

jf

xj j+1j-1

U

( ) ( )

+−+∆−= −+

+ nj

nj

nj

nj

nj

nj ffUffU

htff 11

1

22

( )nj

nj

nj

nj UfUf

htff 1

1−

+ −∆−=

[ ]nj

nj

nj UfUf

htf 112 −+ −∆−=


1+jf

Consider the following initial conditions:

1

1−jf jf

0.1)(2 12/1 =+= −−

nj

nj

nj ffUF 5.0)(

2 12/1 =+= ++nj

nj

nj ffUF

25.1)15.0(5.00.1)( 2/12/11 =−−=−∆−= −++ n

jnj

nj

nj FF

htff

Central Differencing and Stability

25.0)5.00(5.00)( 2/12/311

1 =−−=−∆−= +++++

nj

nj

nj

nj FF

htff

5.0

0.1

=∆=

ht

U


1+jf

Next time step (n+2):

1

1−jf jf

125.1)(2

111

12/1 =+= ++

−+−

nj

nj

nj ffUF 75.0)(

21

111

2/1 =+= ++

+++

nj

nj

nj ffUF

3125.1)125.175.0(5.0125.1)( 12/1

12/1

12 =−−=−∆−= +−

++

++ nj

nj

nj

nj FF

htff

Central Differencing and Stability

5.0

0.1

=∆=

ht

U

Cell j will overflow immediately !!!

Computational Fluid Dynamics IPPPPIIIIWWWWBy considering the fluxes, it is easy to see why the centered difference approximation is always unstable.

nj

nj

nj

ni

ni

nj

nj

ffhtUf

UfUfhtff

>∆+=

−∆−=+

2

211

1+jf

1

1−jf jf

nj

nj ff >+1 Always !



U

Upwind Differencing and Stability

1+jf1

1−jf jf


U1+jf

1

1−jf jf

UfF jj ≈+ 2/1Upwind:

Fj−1 / 2 = Ufj−1n = U

Fj+1 / 2 = Ufjn = 0

)( 2/12/11 n

jnj

nj

nj FF

htff −+

+ −∆−=

)( 1nj

nj

nj UfUf

htf −−∆−=

Upwinding iseffectively anaveraging process!


1

Average


Advect

1


1

Average


1

Advect


1

Average


1

Advect


1−jf jf 1+jf

112/1 == −−njj UfF

02/1 ==+njj UfF


1

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

Upwind Differencing: Unstable Case


1+jf

Integration using upwind scheme:

1

1−jf jf

Fj−1 / 2 = Ufj−1n = U

Fj+1 / 2 = Ufjn = 0

5.1)10(5.10)( 2/12/11 =−−=−∆−= −++ n

jnj

nj

nj FF

htff

1=U 5.1=∆ht

>∆ 1

htU


1+jf1

1−jf jf

Fj−1 / 2 = Ufj−1n =U Fj+1 / 2 = Ufj

n = 1.5U

75.0)15.1(5.10)( 2/12/11 =−−=−∆−= −++ n

jnj

nj

nj FF

htff

25.2)5.10(5.10)( 2/12/311

1 =−−=−∆−= +++++

nj

nj

nj

nj FF

htff


1+jf1

1−jf jf

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


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).


Consideration of Modified Equations:

Why is upwind scheme stable?(Ref: Tannehill et al., Ch. 4)

Computational Fluid Dynamics IPPPPIIIIWWWWDerive modified equation for upwind difference method:

0)( 1

1

=−+∆−

−

+nj

nj

nj

nj ff

hU

tff

Using Taylor expansion:

!+∆∂∂+∆

∂∂+∆

∂∂+=+

62

3

3

32

2

21 t

tft

tft

tfff n

jnj

!+∂∂−

∂∂+

∂∂−=− 62

3

3

32

2

2

1

hxfh

xfh

xfff n

jnj

Computational Fluid Dynamics IPPPPIIIIWWWWSubstituting

−

+∆

∂∂+∆

∂∂+∆

∂∂+

∆nj

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,

!+−∆−+∆−=∂∂+

∂∂

xxxtttxxtt fUhftfUhftxfU

tf

6622

22

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

Computational Fluid Dynamics IPPPPIIIIWWWWTaking further derivatives:

!+−∆−+∆−=+ xxxtttttxxttttxttt fUhftfUhftUff6622

22

!++∆+−∆=−− xxxxtttxxxxttxxxtx fhUftUfhUftUfUUf6622

22222+

+−∆+

∆++−∆+=

)(22

)(22

2

2

hOuUuUx

tOfUftfUf

xxxxxt

ttxttt

xxtt

Computational Fluid Dynamics IPPPPIIIIWWWWSimilarly, we get

),(3 htOfUf xxxttt ∆+−=

),(2 htOfUf xxxttx ∆+=

),( htOUff xxxxxt ∆+−=

Final form of the modified equation:

( ) ( ) xxxxx fUhfUhxfU

tf 132

61

22

2

+−−−=∂∂+

∂∂ λλλ

[ ]3223 ,,, tththhO ∆∆∆+

htU∆=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<∆=h

tUλ

ensures a positive diffusion coefficient

Dissipation( ) xxf!

Dispersion( ) xxxf!

Computational Fluid Dynamics IPPPPIIIIWWWWDissipation vs. Dispersion

Exact Dissipative Dispersive

The nature of the numerical scheme depends on thenature of the lowest order truncation error term.

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 −+−∆−= +

−−

++


Or, substituting

; General representation of various flux formula

)2(2

)(2 1111

1 nj

nj

nj

nj

nj

nj

nj fff

htU

ffhtUff −+−+

+ +−∆

+−∆−=

−+ UU ,

central difference + artificial viscosity

=

2hU

α


While the first-order upwind scheme was found to bestable, it is in general too dissipative (smoothes out all the steep gradients).

Stable and accurate methods:- Lax-Wendroff (I and II)- Leapfrog- Lax-Friedrichs- MacCormack- 2nd order upwind- etc., etc., …

Computational Fluid Dynamics IPPPPIIIIWWWW1. 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

=−+∆− +

−++

+nj

nj

nj

nj ff

hU

tff

nj

nj

nj

nj f

tf

hUf

tf

hU

∆=−

∆+ +

−++

+

12

12

11

111

jnjj

njj

njj Cfbfdfa =++ +

−++

+1

111

1

Implicit Method

Computational Fluid Dynamics IPPPPIIIIWWWWThomas Algorithm

⋅⋅⋅

=

⋅⋅⋅

⋅⋅

⋅⋅⋅⋅⋅⋅

⋅⋅⋅

−

+

+−

+

+

−−−

M

M

nM

nM

n

n

MM

MMM

CC

CC

ff

ff

dbadb

adbadb

ad

1

2

1

1

11

12

11

111

333

222

11

000

0

00

jnjj

njj

njj Cfbfdfa =++ +

−++

+1

111

1

Implicit Method

Computational Fluid Dynamics IPPPPIIIIWWWWThomas Algorithm – The Algorithm

Mjadb

dd jj

jjj ,,3,21

1

!=−= −−

MjCdb

CC jj

jjj ,,3,21

1

!=−= −−

1,,2,11

11 !−−=−=+++ MMk

dfaCf

k

nkkkn

j

Forward Sweep:

Backward Sweep:

Implicit Method


( ) 2/11nj

nj

nj fff −+ +→

2. Lax (Lax-Friedrichs) Method

The forward Euler method can be made stable by

Modified equation

( ) ( )nj

nj

nj

nj

nj ff

htUfff 1111

1

221

−+−++ −∆−+=

( ) !+−+

−=

∂∂+

∂∂

xxxxx fUhfUhxfU

tf 2

2

13

12

λλλ

- Stable for 1<λ- Not uniformly consistent - Still 1st order (dissipative)

( )htU /∆=λ

Lax Method


)(2

211

tOtff

tf n

jnj ∆+

∆−

=∂∂ −+

3. Leap Frog Method

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

Modified equation

( )nj

nj

nj

nj ff

htUff 11

11−+

−+ −∆−=

( ) !+−=∂∂+

∂∂

xxxfUhxfU

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)

Leap Frog Method


!+∆∂∂+∆

∂∂+∆

∂∂+=∆+

62)()(

3

3

32

2

2 ttft

tft

tftfttf

4. Lax-Wendroff’s 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

∂∂=

∂∂

∂∂−=

∂∂

∂∂−=

∂∂

∂∂=

∂∂

LW-I Method


)(2

)()( 32

2

22 tOt

xfUt

xfUtfttf ∆+∆

∂∂+∆

∂∂−=∆+

Substituting

Using central differences for the spatial derivatives

( ) ( )nj

nj

nj

nj

nj

nj

nj fff

htUff

htUff 112

22

111 2

22 −+−++ +−∆+−∆−=

2nd order accurate in space and time

Stable for 1<∆h

tU

LW-I Method

Computational Fluid Dynamics IPPPPIIIIWWWW5. Two-Step Lax-Wendroff’s Method (LW-II)

LW-I into two steps:

For the linear equations, LW-II is identical to LW-I (prove it!)

02/

2/)( 112/1

2/1 =−

+∆

+− ++++

hff

Ut

fff nj

nj

nj

nj

nj

02/1

2/12/1

2/11

=−

+∆− +

−++

+

hff

Ut

ff nj

nj

nj

nj

Step 1 (Lax)

Step 2 (Leapfrog)

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

LW-II Method

Computational Fluid Dynamics IPPPPIIIIWWWW6. MacCormack Method

Similar to LW-II, without

( )nj

nj

nj

tj ff

htUff −∆−= +1

( )

−∆−+= −

+ 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 areidentical to LW-I.

2/1,2/1 −+ jj

MacCormack Method

Computational Fluid Dynamics IPPPPIIIIWWWW7. Second-Order Upwind Method

Warming and Beam (1975) – Upwind for both steps

( )nj

nj

nj

tj ff

htUff 1−−∆−=

( ) ( )

+−∆−−∆−+= −−−

+ nj

nj

nj

tj

tj

tj

nj

nj fff

htUff

htUfff 211

1 221

Predictor Corrector

Combining the two:

( ) ( )nj

nj

nj

nj

nj

nj

nj fffffff 211

1 2)1(21

−−−+ +−−+−−= λλλ

2nd Order Upwind Method

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

20 ≤≤ λ


And the list goes on…


Conditionally consistent

Stable for

Lax-Friedrichs

UnconditionallyStable

Implicit

Stable forUpwind

Unconditionally Unstable

FTCS

0=+ xt Uff

02

111

=−

+∆− −+

+

hff

Ut

ff nj

nj

nj

nj ( ) xxxxx fUhfUt 2

22

2162

λ+−∆−

011

=−

+∆− −

+

hff

Ut

ff nj

nj

nj

nj

( )

( ) xxx

xx

fUh

fUh

1326

12

22

+−−

−

λλ

λ

1≤λ

( ) xxxxx fUhfUh 22

13

12

λλλ

−+

−

( )0

2

11

11

1

=−

+∆− +

−++

+

hff

Ut

ff nj

nj

nj

nj

xxxxx ftUUhftU

∆+−∆ 232

2

31

61

2

( )

( )0

2

2/

11

111

=−

+

∆+−

−+

−++

hff

U

tfff

nj

nj

nj

nj

nj

1≤λ


Stable forSame as LW-I

MacCormack

Stable forSame as LW-I

Lax-Wendroff II

Stable forLax-Wendroff I

Stable forLeap Frog

0=+ xt Uff

022

1111

=−

+∆− −+

−+

hff

Utff n

jnj

nj

nj ( ) xxxfUh 1

62

2

−λ

( )

( ) xxxx

xxx

fUh

fUh

23

22

18

16

λλ

λ

−−

−−

1≤λ

1≤λ

1≤λ

( )

( )0

22

2

2

1122

111

=+−

∆−

−+

∆−

−+

−++

hfff

tU

hff

Ut

ff

nj

nj

nj

nj

nj

nj

nj

02/

2/)( 112/1

2/1 =−

+∆

+− ++++

hff

Ut

fff nj

nj

nj

nj

nj

02/1

2/12/1

2/11

=−

+∆− +

−++

+

hff

Ut

ff nj

nj

nj

nj 1≤λ

( )01 =

−+

∆− +

hff

Utff n

jnj

nj

tj

( ) ( )0

2/ 11

=−

+∆+− −

+

hff

Ut

fff tj

tj

tj

nj

nj


Closing Remarks


Summary by CFD School A

“In solving inviscid flow equations as found in manygas dynamic applications, central differencing scheme is inherently unstable and thus cannot be used. One should use more robust methods such as upwind or other higher order methods in order to ensure stability and accuracy. In general, central differencing scheme is a deficient method in capturing true physical behaviorand should be avoided if at all possible.”


Summary by CFD School B

“Upwind-type schemes applied to the Navier-Stokes equations inherently introduce numerical dissipation which depends on numerical parameters, not on actual physical processes. Sometimes these uncontrolled numerical dissipation may interfere withphysical solution, thereby degrading the fidelity ofsimulation. Central differencing does not suffer fromartificial dissipation and thus preferred as an accuratenumerical method.”

Documents

WPPII Computational Fluid Dynamics I - Unicampphoenics/SITE_PHOENICS/Apostilas/CFD-1_U... · WPPII Computational Fluid Dynamics I Numerical Methods for Wave Equations: Part I: Smooth