time reversed algorithm for pure convection v.М.goloviznin
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Time reversed algorithm for pure Time reversed algorithm for pure convectionconvection
V.М.Goloviznin
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Mathematical modeling transport phenomena on computational grid is one of the fundamental problems of the modern computational mathematics
Simplest transport equation is time reversed
0; 0c c Constt x
Substitution * *; ;t t c c is led to the same equation
* **
0; 0c c Constt x
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Finite difference schemes of time reversed quality
On the regular computational grid in the plane (t,x) are known only two explicit finite difference schemes of second order of accuracy and implicit one.
1 11 1 0;
2 2
n n n ni i i ic
h
x
t
One of them is well knownLeap-Frog scheme
Next one is Iserles scheme
1 11 1 11
0; 2
n n n n n ni i i i i ic
h
x
t
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Finite difference schemes of time reversed quality
Implicit time reversible scheme is also well – known Sn Karlsons scheme
x
t
1 1 1 11 1 1 11
0; 2 2
n n n n n n n ni i i i i i i ic
h h
Leap-Frog scheme is transformed into Arakawa – LillySchemes in multidimensional cases and successfullyExplored in ocean modeling.
Sn – Karlson scheme in the form of dSn-scheme is usedIn neutrons transport calculation for nuclear reactor.
Explicit Iserles scheme is transformed into “CABARET” schemes, witch have a wide sphere of usability.
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“CABARET” schemeIserles scheme can be rewrite as
1/2 1/2
1/2 11/2 1/2 11/2 1
10;
2
n n n nn n ni i i ii i ic
h
Variables will be called as “fluxes variables”.
Variables will be noted as “conservative values”
ni
x
t
1/21/2
ni
The next step of transform gives “two layers form”
1/21/2 1/2 1
1 1/21/2 1
1 1/2 1 11/2 1/2 1
0;2
2 ;
0;2
n n n ni i i i
n n ni i i
n n n ni i i i
ch
ch
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Dissipation and dispersive surfaces
«CABARET» «Leap-Frog»
Dis
sipa
tion
Dis
pers
ion
0.2
0.4
0.6
0.8
-2
0
2
0
2
4
6
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
-2
0
2
0
0.5
1
1.5
2
0
0.2
0.4
0.6
0.8
0
0.25
0.5
0.75
1
-2
0
2
0
0.5
1
1.5
2
0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
-2
0
2
00.25
0.5
0.75
1
0
0.25
0.5
0.75
1
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Since the CABARET scheme is second-order, according to the Godunov theorem it needs some procedure to enforcing monotonicity
i i 1 n
i i 1 n
nn 1 n n n
i 1 i 1 i 1 / 2 ii 1 / 2x x ,x ,t t
nn 1 n n n
i 1 i 1 i 1 / 2 ii 1 / 2x x ,x ,t t
max max max , ,
min min min , ,
We constrain the solution so that
n 1
i 1 i i 1 n
n 1
i 1 i i 1 n
max , x x , x , t t
min , x x , x , t t
Maximum principle
n 1 n 1/ 2 ni 1 i 1/ 2 i2
Consider 3 values inside 1 cell
n n ni i 1/ 2 i 1
n 1i 1Adds on just enough dissipation needed for draining the
energy from unresolved scales, “entropy” condition
New principle item: direct application of maximum principle
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Main distinguishes CABARETfrom upwind leapfrog scheme
CABARET is presented in form of conservation law CABARET has two type of variables : conservative-type and flux-
type CABARET is two-layers scheme with very compact, one-cell-one-
time-level stencil CABARET is monotonic due to direct application of maximum
principle for flux restriction
I+1I
n+1
n
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Explicit Stable under 0<CFL<1/d, d=problem dimension Exact at CFL=0.5, CFL=1 Second-order on arbitrary non-uniform spatial and
temporal grids Conservative Satisfies a quadratic conservation law Non-dissipative Very compact, one-cell-one-time-level stencil Small dispersion error Direct application of maximum principle for flux
restriction No adjustment parameters
Main features of the CABARET scheme
I+1I
n+1
n
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Another reason to call it CABARET…
Computational stencil of the forerunner of CABARET scheme
“Compact Accurately Boundary Adjusting high-REsolution Technique for Fluid Dynamics
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CABARET for gas dynamics flows.First unique feature.
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Gas dynamics: verification test Contact discontinuity
Плотность, CFL=0.5, NT=150
0,9981
1,0021,0041,0061,0081,01
1,012
1 21 41 61 81 101
Плотность, CFL=0.05, NT=1500
0,9981
1,0021,0041,0061,0081,01
1,012
1 21 41 61 81 101
Плотность, CFL=0.5, NT=1500
0
200
400
600
800
1000
1200
1 21 41 61 81 101
Плотность, CFL=0.05, NT=1500
0
200
400
600
800
1000
1200
1 21 41 61 81 101
21, 10L M R L M Ru u u p p p Weak contact discontinuity
Strong contact discontinuity
independence from amplitude
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independence shock wave thickness from amplitude
0.91
1.1
1.21.31.41.5
1.61.7
0 10 20 30 40 50 60 70
давл
ение
0.999995
1
1.000005
1.00001
1.000015
1.00002
0 10 20 30 40 50 60 70
давл
ение
0.00E+00
2.00E+05
4.00E+05
6.00E+05
8.00E+05
1.00E+06
0 10 20 30 40 50 60 70
давл
ение
Very slow shock wave
Very strong shock wave
Ordinary shock wave
510P
P
610P
P
1P
P
First Unusual Feature of CABARET:
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Verification task Blast Wave problem
0.0 0.1
, 0.1 0.9
0.9 1.0
L
M
R
if x
x t if x
if x
2 23
, , ,
1,
0,
10 , 10 , 10 .
T
L M R
L M R
L M R
u p
u u u
p p p
0
1
2
3
4
5
6
7
0 50 100 150 200
N
пот
но
сть
0
1
2
3
4
5
6
7
0 500 1000
N
пл
отн
ост
ь
P,Woodward, P,Colella J,Comp,Phys,, 54, 115-173 (1984)
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1-D shock interaction with density perturbations: Shu&Osher problem
0
1
2
3
4
5
0 0.2 0.4 0.6 0.8 1
x/L
200
Converged
0
1
2
3
4
5
0 0.2 0.4 0.6 0.8 1
x/L
400
Converged
Shock capturing capability without notable dissipation
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Double Mach reflection test
Grid (481x121)Grid (961x241)Grid (1921x481)
In a semi-open domain an oblique shock wave of Mach equal to 10 impinges on the horizontal reflective boundary under an angle of 600
Titarev and Toro, 2002; J.Qiu and C.-W. Shu, 2003
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CABARET for aeroacoustics problems.Second unique feature.
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D2 acoustic Gaussian pulse propagation on nonuniform grid
Initial condition
0 0
0
, , ( , )
, 0
( , ) 1/ , ( , ) 1, ( , ) 0
t t
t
fon fon fon
x y p x y x y
u x y
p x y x y u x y
2 20 0( , ) exp ( )p x y x x y y
5
0 0
( , ) 100,100 100,100 ,
( , ) (0,0), 10 , log 2 / 9
x y
x y
Computational grid:
Second Unusual feature of CABARET:
Acoustic disturbances is not dissipate
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D2 acoustic Gaussian pulse propagation on nonuniform grid
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Simulation of vortex flow.Third unique feature.
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2-D zero-circulation compressible isentropic vortex in a periodic box
H=1
L=0.05 One revolution: T=1.047
1
12 2 2
12 2 2
2 2 10 0 0 0
( 1)' 1 exp{2 (1 )} 1 , ' exp{ (1 )}sin ,
4
( 1)' exp{ (1 )}cos , ' 1 exp{2 (1 )} 1 ,
4
/ ; ( ) ( ) ; tan (( ) /( )),
u
v p p
r L r x x y y y y x x
Stationary and stable solution to EE. But how long can the numerical scheme hold it?
Full Euler equations are solvedKarabasov and Goloviznin, 2008
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Single Vortex
PresureEntropy
Computational grid 50х50
Third Unusual feature of CABARET:
Stationary vortex is not dissipate
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Vortex Dipole
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Vortex preserving capability: Problem of a steady 2-D zero-circulation compressible vortex in a periodic box domain
t=100
(30x30), 1.5 points per radius (p.p.r.)
(60x60), 3 p.p.r.
(120x120), 6 p.p.r.
Conserves total k.e. within ~ 1%
Vortex in a box: stationary and stable solution to the Euler equations. But how long can the numerical scheme preserve it?
Vorticity
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Vortex preserving capability: what happens with a conventional 2nd-3rd order conservative method? (e.g., Roe-MUSCL-TVD, grid (240x240))
With the TVD limiter: t=4 With the TVD limiter: t=100 No limiter: t=4
With the limiter the solution is too dissipative
Without the limiter it is too dispersive
(240x240)12 points per vortex radius
Vorticity
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Vortex preserving capability & shock-capturing: Zero circulation vortex interaction with a stationary normal shock wave in a wind tunnel: grid (400 x 200), density field shown
Weak vortex
Strong vortex
Zhou and Wei, 2003; Karabasov and Goloviznin, 2007
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D2 Backward Step Re=5000
40 greed point on step
20 greed point on step
10 greed point on step
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CABARET for uncompressible flows.
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Stream instability on grid (256 x 256)
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Flow behind turbulizing grid
Real stream CABARET simulation (256х512)
FLUENT simulation
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Submerged jet on computational grid (128 x 640)
Foto
Result of simulation. Vorticity Field.
Animation
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Towards Empiricism-Free Large Eddy Simulation for Thermo-Hydraulic Problems
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Aanimation
15MC,Re=85000
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Mixing hydrogen under containment
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Remarkable characteristic of CABARET
•independence shock wave thickness from amplitude;•acoustic disturbances is not dissipate.•stationary vortex is not dissipate;
CABARET applicable for lot of challenging problems:•Transonic aerodynamics•Aeroacoustics•Vortex flow simulation•Ocean modeling•Atmospheric pollution transport•Strongly nonuniform reservoir modeling,•Combustion modeling•Computing turbulent fluid dynamics•Et al
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Conclusions
Business problem: implementation of CABARET in the industry
Innovative scientific problem: spreading of CABARET on new sphere of science andincreasing order of accuracy up to fourth.
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Publication
•V.M.Goloviznin “Digital Transport Algorithm for Hyperbolic Equations”/ V.M.Goloviznin and S.A.Karabasov – Hyperbolic Problems. Theory, Numerics and Application. Yokohama Publishers, pp.79-86, 2006
•Goloviznin, V.M. and Karabasov, S.A. New Efficient High-Resolution Method for Nonlinear Problems in Aeroacoustics, AIAA Journal, 2007, vol. 45, no. 12, pp. 2861 – 2871.
•Karabasov S.A., Berlov P.S., Goloviznin V.M. CABARET in the ocean gyres. Ocean Modelling. Ocean Model., 30 (2009), рр. 155–168.
•Goloviznin V.M. CABARET finite-difference schemes for the one-dimensional Euler equations / V.M. Goloviznin, T.P. Hynes and S.A. Karabasov // Mathematical Modelling and Analysis, V.6, N.2 (2001), pp. 210-220
• Goloviznin V.M., Karabasov S.A. Compact Accutately Boundary-Adjusting high-Resolution Technique for fluid dynamics. Journal of Computational Physics, 2009, J. Comput.Phys., 228(2009), pp. 7426–7451.
•V.M.Goloviznin A novel computational method for modelling stochastic advection in heterogeneous media./ Vasilly M. Goloviznin, Vladimir N. Semenov, Ivan A. Korotkin and Sergey A. Karabasov - Transport in Porous Media, Volume 66, Number 3 / February, 2007, pp. 439-456
•Goloviznin V.M. Direct numerical modeling of stochastic radionuclide advection in highly non-uniform media / V.M. Goloviznin, Kondratenko P.S., Matweev L.V., Semenov V.N., Korotkin I.A. – (Preprint IBRAE № IBRAE –2005-01)- М.: ИБРАЭ РАН, 2005, -37 p.
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Thank you for your attention