Euler Equation by Jameson Method

CFD Lecture 18-20Euler Equations

Jameson Finite Volume Scheme

Prof. Ken Gordon

Solve Euler Equations over 2-D domain using Jameson Finite Volume Scheme:

1. Euler Equations• Classification• Conservative (differential) form (5.5)• Transformation to computational () domain (5.6)

2. Finite Volume (FV) vs. Finite Difference (FD) schemes (3.5, 5.7)• Development of FV form: cell volume vs. fluxes• General discretization with (small) grid cell• Three steps:

a) Gridding: Cell-centered vs. Nodal-pointb) Volume / Fluxes: select scheme to approximate integrals (6.3-6.5)c) Time step: select scheme to update cell-averaged parameters

3. Boundary Conditions (6.7)• Walls: Dummy-cell, extrapolation• Far-field: Characteristics (information flow) (6.2)

– Linear, Non-Linear (Riemann Invariants)

Overview

Class 18

Class 19


4. Smoothing• “Artificial viscosity” vs. Inherent numerical dissipation

5. Stability Analysis• CFL condition, max local t

6. Initial Conditions

Write & Run your code! (steady-state solution, not time accurate)

Overview

Class 20

Recall equations in 2-D are:

Euler Equations (Inviscid, Adiabatic Flow): Overview

0

yF

xE

tU

tevu

U

upeuvpu

u

E

t

2

vpepv

uvv

F

t

2where

22221 wvuTce

RTp

t

and thermodynamic / state equations for ideal gas:

22221 wvuTc

peh

p

tt

Subsonic(M < 1)

Sonic(M = 1)

Supersonic(M = 1)

EllipticHyperbolic

ParabolicHyperbolic

Hyperbolic (in space)Hyperbolic (in time)

Steady Flow:Unsteady Flow:

These are Non-Linear system of equations in Strong Conservative formOften appropriate for predicting core velocity and pressure field – use for initial designs

Classification:

Examples of Elliptic, Hyperbolic steady-flow fields (body in flow)

Euler Equations: Classification

What happens if … Flow is subsonic?

Flow is supersonic?

Flow

Wind tunnel

Convert strong-conservative differential form to pair of volume / surface integrals:

Finite Volume Method: Volume / Flux integrals

0

dAyF

xE

tU

A

Green’s Theorem converts a volume (area) to surface integral

0

SA

dxFdyEdydxUt

SA

dSnBdAB

For 2-D flow, assume unit depth (in z), so volume integral is ~ area integral

Apply using and , integrating CCW:

A

dAyF

xE

jFiEB

jdSdxi

dSdyn

Change of U within area

Fluxes E and F crossing surfaces

(CCW integral)

nx = cosdy = dS cos

S

A

dS

-dx

nx

nny

|n|=1

Euler equations become:

SS

FdxEdydSdSdxF

dSdyE

We want to discretize this conservative integral equation over small Control Volumes.

Finite Volume Method: Small Control Volume (Area)

0

SA

dxFdyEdydxUt

Simple example: Consider a small rectangular area

x

y EE

FN

EW

FS

0

dxFdyEdxFdyEyxtU

SWNEU

xFyExFyEyxt

USWNE

1

Three-step plan for Finite Volume process:a) Gridding: Discretized points to store values of fluid variablesb) Volume / fluxes: select scheme to approximate area & surface integralsc) Time step: select scheme to update cell-averaged parameters

For example, consider first row of U, E, F in Euler equations (mass conservation):

xvyuxvyuyxt SWNE

1

Integrationdirection

(CCW)

Advantages to FINITE VOLUME vs. FINITE DIFFERENCE formuation:

Finite Volume Method: Why??!!

0

SA

dxFdyEdydxUt

0

dAyF

xE

tU

A

a) Conservative discretization: locally and globallyMass, momentum, energy is conserved between adjacent cells.Conservation maintained over entire domain since interior surface integrals

cancel.b) Can work directly in (x,y) domain.

No need to transform to (,) or use metrics.

Two grid methodologies: Nodal-Point vs. Cell-Centered

Finite Volume Method: Grid Layout

Nodal-Point (Face-Centered)

Define nodal locations first ( )i,j.

Variables (e.g. , u, v, p) stored at grid nodes.Control volumes connect nodes.

Boundary

NE

SW

Cell-Centered (Node-Centered)

Boundary

NE

SW

Advantage: Nodal values represent mean over CV better since nodes at centroid.

Advantage: Approximation of fluxes better for skewed cells since faces connect computational nodes.

Establish CVs with suitable grid (–).Geometry (x,y) stored at grid intersection points.Assign computational node ( )i,j to CV centroid.

Variables (e.g. , u, v, p) stored at grid nodes.

More typically used

Use cell-centered scheme on Euler equations

Jameson Finite Volume Scheme: Integral Evaluations 0

SA

dxFdyEdydxUt

1) Volume integral:

NE

SW i,j

i+1, j

i-1, j

i, j+1

i, j-1

a

b

d

c

tU

AdydxUt

jiji

A

,

,

Assume Ui,j represents value over CV

bdacbdacji xxyyyyxxA 21

,

For area Ai,j calculated as

Control volume (i,j) has area Ai,j, nodes (a,b,c,d), and faces (E,N,W,S).

Make approximation that each face is a straight line.

Faces at: E (i+½, j), W (i-½, j), N (i, j+½), S (i, j-½)

Use cell-centered scheme on Euler equations

Jameson Finite Volume Scheme: Integral Evaluations 0

SA

dxFdyEdydxUt

2) Surface integrals:

NE

SW i,j

i+1, j

i-1, j

i, j+1

i, j-1

a

b

d

c

Two choices for flux at (say) East face:i) FE = F(UE) for UE = ½ (Ui,j+ Ui+1,j)

ii) FE = ½ (Fi,j+ Fi+1,j)

Method (ii) is chosen in general, but (i) can be just as good.

xFyEdxFdyES

Also, fluxes E and F can be evaluated at time level k (explicit) or k+1 (implicit).Jameson is explicit scheme. Fluxes evaluated using known values from last completed time step k.

EENNEE xFyEyE

Where (CCW!),, bcEbcE xxxyyy ,, daWdaW xxxyyy

3) Combine for Basic Jameson Scheme

0,,

SSNNWWEESSNNWWEEji

ji xFxFxFxFyEyEyEyEtU

A

Work directly in (x,y) domain!

Transform equations from physical (x,y) to computational (,) domain

Jameson FV Scheme: Comparison to Finite Differencing

yF

xE

tU

xyxy

Jyx

yx 1

0

xFxFyEyEtUJ

Return to Strong Conservative form: 0

t

0

FxEyFxEyFxEyFxEytJU

Better than textbook: uses metrics in form we calculate them

Use values of metrics:

0ˆˆˆ

FE

tUCan think of this like Can then apply finite differencing

schemes over 2-D grid (,).

0

yyxx FFEEtU

Finite difference with central schemes:

Jameson FV Scheme: Comparison to Finite Differencing

ddx

ddy

ddy

ddx

J

xyyxxyyx

OCOBOCOBA

CBCB

sinProof: Area of quadrilateral:

0

FxEyFxEytJU

tU

JtJU ji

ji

,,

1) Time derivative term:

Jacobian transforms area of computational element to true FV area

x

y

A

tUA jiji

,,

Finite difference with central schemes:

Jameson FV Scheme: Comparison to Finite Differencing 0

FxEyFxEytJU

jijijiji yEyEEy ,,,,

21

21

21

21

jijiji EEE ,1,21

,21

2) One spatial term: Use central differencing about mid-point

i,j i+1, jwhere we can define values at ½ plane:

WWEEadWbcE yEyEyyEyyE

11

a b

d c

1121

21

21

21

21

21

21

21

21

21 ,,,,,, jijijijijiji yyEyyEEy

3) Combine time derivative and all four flux terms for:

0,,

SSNNWWEESSNNWWEEji


A

This is exactly Jameson’s (basic) Finite Volume scheme!

Nodal-Point scheme has better flux accuracy for skewed gridding

Nodal-Point FV Scheme: Fluxes

2) Surface integrals:

NE

SW i,j

i+1, j

i-1, j

i, j+1

i, j-1

a

b

d

c

Two choices for flux at (say) East face:i) FE = F(UE) for UE = ½ (Ui,j+ Ui+1,j)

ii) FE = ½ (Fi,j+ Fi+1,j)

Method (ii) is chosen in general, but (i) can be just as good.

xFyEdxFdyES

Also, fluxes E and F can be evaluated at time level k (explicit) or k+1 (implicit).Jameson is explicit scheme. Fluxes evaluated using known values from last completed time step k.

EENNEE xFyEyE

Where (CCW!),, bcEbcE xxxyyy ,, daWdaW xxxyyy

3) Combine for Basic Jameson Scheme

0,,

SSNNWWEESSNNWWEEji


A

Work directly in (x,y) domain!

THIS SLIDE IS TEMPORARILY BOGUS!!


1. Euler Equations• Classification• Conservative (differential) form (5.5)• Transformation to computational () domain (5.6)

2. Finite Volume (FV) vs. Finite Difference (FD) schemes (3.5, 5.7)• Development of FV form: cell volume vs. fluxes• General discretization with (small) grid cell• Three steps:

a) Gridding: Cell-centered vs. Nodal-pointb) Volume / Fluxes: select scheme to approximate integrals (6.3-6.5)c) Time step: select scheme to update cell-averaged parameters



Overview

Class 18

Class 19

Many hyperbolic methods to advance from one time step to next.

Consider conceptual differential equation Goal: Would like explicit, at least O(t2) error scheme

Time Step Evaluation

1. Leap-frog method:Multi-Step scheme requires special starting procedures from U0 to U1 before

getting to U2 and later times.

)(UfdtdU

)( 111 kkk UftUU

2. Runge-Kutta (Single-step, Multi-stage scheme)No special starting requirements. Advance directly from Uk to Uk+1, but may take

additional calls of flux terms in-between.“The” 4-stage R-K is generally written for as

tKKKKUU kk 4321611 22

),( tUfdtdU

),(

),(

),(

),(

134

221

3

121

2

1

21

21

kk

kk

kk

kk

ttKUfK

ttKUfK

ttKUfK

tUfK

Jameson-Schmidt-Turkel high-order error, lower-storage, 4-stage scheme:

Jameson FV Scheme: Time Step Evaluation

O(t2) scheme, stable for .22

xta

1,

)3()0()4(

)2(21)0()3(

)1(31)0()2(

)0(41)0()1(

kjiURtUU

RtUU

RtUU

RtUU

0,,

fluxestU

A jiji ji

ji

ji RfluxesAt

U,

,

, 1

Original Jameson uses fact that RHS is not explicitly dependent on time.

kjiU ,

kjiUU ,

)0( )0(R )0(U

1

,)2()0(

21)0()3(

)1()0(21)0()2(

)0()0()1(

kjiURRtUU

RRtUU

RtUUO(t2) scheme, stable for .

Requires extra storage of residuals.

2

xta

Consider current state , and define and as residual based on .

Rewrite as

For SS problems can think of R as a residual, and want to drive R → 0.

Solid Walls (or known streamlines / symmetry)

Boundary Conditions: Solid Walls

0

SA

dxFdyEdydxUt

Typical code(# lines)

PDE: 30-35%BC: 50-55%Smoothing: 15%

i,1

a

bHow to calculate flux on ab wall?

b

ayx

b

a

dSnFnEdxFdyE

Given along solid boundary / streamline: 0 yx nvnunu

Euler equations:

dSn

vhpv

uvv

n

uhuvpu

ub

ay

t

x

t

2

2

dS

hnupnvnupnunu

nub

a

t

y

x

0

0

0

0

dxp

dypdS

pnpnb

a y

x

Makes sense physically! No mass / enthalpy flux through across streamline.

Only momentum flux due to pressure force on wall.

Jameson implementation: Approximate Pwall over Pa to Pb :

Jameson FV Scheme: Solid Walls

(i,1)

a

b

ii) 2nd-order extrapolation:

But this gives O(x) error!Degrades entire solution to 1st -order accuracy.

i) Simple extrapolation: Pw = P1.

Pw

P1

xPyP

dxp

dyp

w

w

0

0

1

1

xPyP

dxFdyEb

a

nRvuPPw

22

1

Rvu

nPP

nP w

221

So, for curvature (a),

nRvuPPw

22

1

and for curvature (b).

Use (u,v) from point (i,1), and n & radius of curvature R from boundary grid points.

a

bPhigh

Plow(b)

a

bPlow

Phigh

n

(a)

iii) Dummy (Ghost) cell reflection (recommended):

Jameson FV Scheme: Solid Walls

b

a

dxFdyECalculate as usual, like an interior point.

10

10

pp

10

10

tutununu

Set and → Normal opposite sign

→ Tangential same sign

So, when averaging E and F on face ab, get mass flux = 0

Problem with Nodal-Point scheme:

0

0

21

xy

PPdxFdyE ba

b

a

Leads to Phigh/Plow oscillation, driving fluctuating velocity vectors.

Can add “transpiration” termsto correct for flow tangency:

yorigc

xorigc

nnuvv

nnuuu

but then mass flux is not conserved.

a b

(i,1)

a

b

(i,0)


1. Euler Equations

2. Finite Volume (FV) vs. Finite Difference (FD) schemes (3.5, 5.7)



4. Smoothing• “Artificial viscosity” vs. Inherent numerical dissipation

5. Stability Analysis• CFL condition, max local t

Write & Run your code! (steady-state solution, not time accurate)

Overview

Class 20

Trade-off: Accuracy vs. Size of Domain

Boundary Conditions: Far-Field

Far from body, u → U∞, v → 0, streamlines are parallel so p → p∞.

1.Can set u = U∞, v = 0. Accurate to 0.1% by ~32 radii (1/r2) to 1000 radii (1/r)

1. Where to put boundaries2. How to model behavior3. How many BCs to apply, and which ones?

upstream

U∞ radius, a

222 yxyUu

222 yx

xv

2

log1rr

r

r1

1cos21 22

2

ra

Uu

Recall for circle,

UG potential theory (source/sinks):

Circulation (lift) ~

(drag) ~

Doublet (thickness) ~ 21 r

General shape

2. For cases with circulation, set correction ,

For 0.1% accuracy, can reduce domain back to ~32 radii.

Careful not to over-constrain problem

Consider 1-D incompressible (subsonic) steady nozzle flow:


Steady subsonic flow is elliptic, could specify u and P at either location.Full unsteady Euler equations are elliptic/hyperbolic …

1. Where to put boundaries2. How to model behavior3. How many BCs to apply, and which ones?

P0, u0

P1, u1

How many boundary conditions need be specified?

constuA Specify: u0 (inlet velocity)

Specify: P1 (back pressure)

constuPdxdP

dxduu 2

2101

Improved BC methodology: (Non-linear) Riemann InvariantsRewrite Euler equations under assumption no shocks, :


0

0

01

011

xsu

ts

xvu

tv

xp

xuu

tu

xu

xu

t

1dpdedsT

s

consts

epa 12

sep

0 y

Ideal gas: Tce Supporting Equations:

RTp Entropy:

Eqn of state:

Speed of sound:

Thermo: ccR p ccp

dsadad

dsdada

sa

11

12

12

ln1lnln2

Already in convective form

Put in convectiveform

dsdpdp

dsadaadp

112 2

(substitute in mass conservation)

Also,

(substitute in x-momentum eqn)

(Non-linear) Riemann Invariants:


0

0000000000

)1(

)1(2

2

JJsv

xau

auu

u

JJsv

ta

a

auJ1

2

011

2

01

21

2

2

xsa

xaa

xuu

tu

xua

xau

ta

0

0

01

011

xsu

ts

xvu

tv

xp

xuu

tu

xu

xu

t

In matrix form:

A

B

A B =

011

21

2 2

xsaau

xauau

t

Now, ignore off-diagonal terms with assumption of no entropy gradients (esp. no shocks)

where

Riemann Invariants

specify v, s, J+ at (1), J- at (2)

specify v, s, J+, J- at (1), nothing at (2)

(Non-linear) Riemann Invariants: v, s, J+, J- where , and

See:

Boundary Conditions: Far-FieldauJ

12

So at each boundary, need to specify from outside domain:

v, s advected at velocity uJ+ advected at velocity u+a

J- advected at velocity u-a

u > a must specify v, s, J+, J-

0 < u < a v, s, J+

-a < u < 0 J+

u < -a nothing

For example:• If flow is subsonic,• If flow is supersonic,

pes

pa 2

u > 0

(1) (2)

For example:

Implementation: Dummy (Ghost) cell reflection (recommended):

Jameson Boundary Conditions: Far-Field

• Apply to each cell, based on local u and a conditions (and use p/ rather than s)• Changing U for each iteration may change # BCs that need be applied

• For top & bottom boundaries, rotate coordinate system

(1,j)(0,j)

Calculate fluxes for cell (1,j) as usual, like an interior point.

Calculate from v, s, J+, J- in right direction.

e.g. for subsonic inflow, jU ,0

ffjoffjoffjo JJssvv ,,, ,, (coming in from far-field)

jjo JJ ,1, (coming out from inside)

(N,j) (N+1,j) e.g. at other end of domain, for subsonic outflow,

jNjNjN Jsv ,1,1,1 ,, set by at (N, j)

ffjo JJ , (coming in from far-field)Jsv ,,

ff

ff

(Linearized) Riemann Invariants – TheoryRewrite Euler equations using primitive variables under assumption :

Boundary Conditions: Far-Field (Linearized Approach)

0

xUA

tU

0 y

pvu

U

using .

Linearize about fixed A matrix: pppvvvuuu ~,~,~,~

0

0

01

0

xsu

ts

xvu

tv

xp

xuu

tu

xu

xu

t

0~~

0~~

0~1~~

0~~~

xsu

ts

xvu

tv

xp

xuu

tu

xu

xu

t

0~~~

xup

xpu

tp

Some work, using ~~~ d

ppdsd

0

~~~~

00000

10000

~~~~

pvu

xup

uu

u

pvu

t

Therefore,

(matrix form)

Get characteristics of 1-D Euler flow from eigenvalues/vectors of

Boundary Conditions: Far-Field (Linearized)

p

atxV

p

atxVtxVtxV

pvu

0,

0,

0001

,

0100

,

~~~~

4321

upu

uu

A

00000

10000

Set : 03

upuu 0det IA

auu 4,32,1 ,

0222 auu pa 2where

Eigenvalues:

Total solution:

Shearwave

Entropy wave(pure variation)

Isentropic pressurewaves (e.g. acoustic)

)( au )( au )(u )(uConvection speed:

pvu

papaa

VVVV

~~~~

210210210210

10010100

2

4

3

2

1

And inverse transformation:

Apply to (0,j) as

Only outgoing p wave (V4), where from inverse transformation

At far-field, replace with far-field conditions , then set Vi = 0 for incoming waves, extrapolate Vi for outgoing waves to compute .

Boundary Conditions: Far-Field (Linearized)

For example if flow is subsonic throughout, • first three waves travel to right (u, u, u+a)• fourth wave travels to left (u-a) u > 0

(1) (2)

So at inlet (1) assume far-field disturbances are zero V1 = V2 = V3 = 0

ff

ff

ff

ff

ff

ff

ff

jp

aV

pvu

pvu

04

),0(

ppuaV ~21~214

(1,j)(0,j)ff

apvu ,,,, ffffffffff apvu ,,,,pvu ~,~,~,~

Dummy-cell implementation:

ff

ffj

ff

ffj

ppp

auu

V22,1,1

4

Calculate V4 from perturbation at (1,j) as

(similar analysis at exit (2), where V4 = 0, and get V1, V2, V3 from (N,j) )

We saw

Finite-difference schemes typically require “artificial viscosity” smoothing1. High wave # oscillations (order of grid spacing) solutions to FDE but not PDE2. Arise also from non-linearities which can be unstable

Smoothing (Dissipation)

0 xt cuu 02 11

jjj uu

xc

tu jju 1

Recall amplification factor process:

odd/even oscillation

In FD scheme, dissipation is inherent part, but don’t want to rely on it because it’s uncontrollable.

Use: xxxxxxxt uxuxcuu 342

2nd orderviscosity

4th orderviscosity

qtxit eetxu ,

uqiuuu

uuuiu

txxxx

xxx

,

,,4

2

ctxitxx eetxu

cqxx

xxicqi

434

22,

:Im:Re 43

42

2

434

22Substitute for:

x

uuuuux

uuuxuuc

tu jjjjjjjjjjj

2112411211 4642

2

Important notes:• 2, 4 > 0 give stable solution

• 4 targets higher wave # (decays quicker in x)

• values of viscosity independent of grid (all 1/x)

• 2 introduces O(x) error

• 4 introduces O(x3) error

Implementation like:

Use only at shocks / discontinuitiesUse over entire flow field

has solution

Apply dissipation term to RHS of finite volume equation:

Jameson Implementation of Smoothingji

jiji Dfluxes

tU

A ,,

,

UUUSUSD tA

tA

tA

tA

ji33

42,

Where: S, S are switches (turn on/off near high gradients),

A, t are local cell area and time step,typically 2 ~ 0.05, 4 ~ 0.01,

, are differencing operators, e.g.:

jijijiji

jijiji

jiji

jiji

UUUUU

UUUUU

UUU

UUU

,1,,,13

,1,,12

,,

,,

21

21

21

21

21

21

21

21

33

2

For cell-centered finite volume method, use as 0flux modified,,

tU

A jiji

i,j

a

b

d

c

EN

W S

jijijijiE

jijiE

EEEEE

UUUUtAUU

tAS

xFyEge

,1,,1,24,,12 33

flux mod..

See 3rd-order differencing template extends two to the eastEvaluate S, A, t on face (need to average across neighboring cell)

Further implementation notes for specific faces:

Jameson Implementation of Smoothing 0flux modified,

,

t

UA ji

ji

i,j

a

b

d

c

EN

W S

jijijijiW

jijiW

WWWWW

UUUUtAUU

tAS

xFyEge

,2,1,,14,1,2 33

flux mod..

W face: Use 3rd-order difference template extending two to the west

S face: Along solid wall, set smoothing flux = 0 (to ensure conservation)

N face: Will need 3rd-order differencing template two to north (OK) and one below (beyond wall!)1. use dummy cell (i,0) (easy, since already recommended earlier)2. use one-sided difference away from wall.

2nd order viscosity switch: want to turn on only near shocks / discontinuities

Jameson Implementation of Smoothing

Varies from 0 to 1.

Only apply if expect strong enough shock / discontinuity to warrant.

Can also track s (entropy) instead.

1,

)3()0()4(

)2(21)0()3(

)1(31)0()2(

)0(41)0()1(

kjiURtUU

RtUU

RtUU

RtUU

Freeze Di,j smoothing during multi-stage Runge-Kutta time-stepping.

Just evaluate dissipation fluxes once, and use for all inner time stages:

jijiji

jijiji

ppp

pppS

,1,,1

,1,,1

2max

2

1,,1,

1,,1,

2max

2

jijiji

jijiji

ppp

pppS

(max throughout flow field)

ji

jijiji

p

pppS

,

,1,,1 2

ji

jijiji

p

pppS

,

1,,1, 2

or,Adjusts 2 “automatically”, but also a problem in low pressure areas.

1

,)0()3(1)0()4(

)0()2(21)0()3(

)0()1(31)0()2(

)0()0(41)0()1(

kjiA

A

A

A

UDfluxestUU

DfluxestUU

DfluxestUU

DfluxestUU

jiji

ji DfluxestU

A ,,

,

Explicit scheme time steps need to be limited to prevent unstable solution.Approach: Evaluate stability requirement (like CFL) at each cell (i,j) ti,j.

Global time stepping: (for time accurate solution throughout domain) No time step anywhere can be larger than min local:

Local time stepping: (for more rapid iteration to steady-state solution) Use 0.9 ti,j for each cell (i,j)

Stability (Temporal)

jijiglobal tt ,,min9.0

How to find max(ti,j) for the Jameson scheme (system of 2-D, non-linear equations)?

Still use Von Neumann (linearized) stability analysis (ignoring BCs)

1) Stability of 1-D system of equations:

0

xU

tU

xE

tU A

0

~~~~

00000

10000

~~~~

pvu

xu

uu

u

pvu

t

Hold A constant while U is advanced through one t.

Saw linearized Euler equations could be written as:

e.g. Euler:

2) Stability of 1-D system of equations (continued)


Apply for linearized Euler [A] matrix and Lax-Wendroff time stepping scheme:

Define amplification matrix, [G]: kk UGU 1

xit eUtxUtxU ),(),(Consider one wavenumber :

Stability given by maximum eigenvalue of [G] < 1 (not just |G| < 1) Depends on [A] matrixand FD scheme!

kj

kj

kj

kj

kj

kj

kj UUU

xtUU

xtUU 11

22

111 2

21

21

AA

2

22 cos1sin

AA

AA

baI

xxiIG xt

xt

xit eeU

0 xt ucu(Like old wave equation , except c is [A])

Look at development of one wave number:

Solve for amplification matrix:

2) Stability of 1-D Euler equations using Lax-Wendroff scheme (continued)


Can show eigenvalues of G are (1+a+b2) where are eigenvalues of A: auu 4,32,1 ,

Requirement that magnitude of (eigenvalue of G) < 1 implies

1sincos11 2222

xx x

txt

For stability, it can be shown this reduces to

xtxt

1

Which for the worst case givesauxt

VbaVba 22 11 AA

VV AIf V is an eigenvector of A, then

VbaGV 21

Like the CFL condition for 1-D wave equation.

Numerical speed (x / t) must still exceed physical speed (|u|+a)!

3) Stability of 2-D Euler equations for multi-stage time step scheme (Jameson!)


Far too laborious to do entirely here. It involves:

0ˆˆ

FE

tUJ

yF

xE

tU

a) Transforming to () domain

b) Linearize as 0

UU

tU BA

c) Discretize in () (consistent with face fluxes), and apply 4-stage time-step scheme to individual wave number U to get amplification matrix G.

d) Interrogate eigenvalues of G (based on eigenvalues of A and B). For stability require largest magnitude of eigenvalue to be < 1.

222222,

yxyxavuJKt ji

Yields formula like:

Where K has value:

MacCormack 13-stage R-K 24-stage R-K 22

For rectangular mesh, v=0 get:

22,1 yxau

xKt ji

Cells that with higher AR or more skewness also reduce max. local t.

avy

auxKt ji ,min,

Sometimes approximated as:

-5 -4 -3 -2 -1 0 1 2 3 4 5

-3

-2

-1

0

1

2

3

4

5

Mesh WITH adding ghost points

-5 -4 -3 -2 -1 0 1 2 3 4 5

-3

-2

-1

0

1

2

3

4

5

Mesh WITHOUT adding ghost points

Grid:

Homework

Documents

Euler Equation by Jameson Method