Euler Equation by Jameson Method
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Transcript of 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
Solve Euler Equations over 2-D domain using Jameson Finite Volume Scheme:
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
ji xFxFxFxFyEyEyEyEtU
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
ji xFxFxFxFyEyEyEyEtU
A
Work directly in (x,y) domain!
THIS SLIDE IS TEMPORARILY BOGUS!!
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
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)
Solve Euler Equations over 2-D domain using Jameson Finite Volume Scheme:
1. Euler Equations
2. Finite Volume (FV) vs. Finite Difference (FD) schemes (3.5, 5.7)
3. Boundary Conditions (6.7)• Walls: Dummy-cell, extrapolation• Far-field: Characteristics (information flow) (6.2)
– Linear, Non-Linear (Riemann Invariants)
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:
Boundary Conditions: Far-Field
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, :
Boundary Conditions: Far-Field
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:
Boundary Conditions: Far-Field
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)
Stability (Temporal)
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)
Stability (Temporal)
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!)
Stability (Temporal)
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:
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