Time reversed algorithm for pure convection V.М.Goloviznin.

Time reversed algorithm for pure Time reversed algorithm for pure convectionconvection

V.М.Goloviznin

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

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

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.

“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

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

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

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

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

Another reason to call it CABARET…

Computational stencil of the forerunner of CABARET scheme

“Compact Accurately Boundary Adjusting high-REsolution Technique for Fluid Dynamics

CABARET for gas dynamics flows.First unique feature.

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


0,9981

1,0021,0041,0061,0081,01

1,012

1 21 41 61 81 101


0

200

400

600

800

1000

1200

1 21 41 61 81 101


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

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:

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)

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

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

CABARET for aeroacoustics problems.Second unique feature.

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

D2 acoustic Gaussian pulse propagation on nonuniform grid

Simulation of vortex flow.Third unique feature.

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

Single Vortex

PresureEntropy

Computational grid 50х50

Third Unusual feature of CABARET:

Stationary vortex is not dissipate

Vortex Dipole

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

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

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

D2 Backward Step Re=5000

40 greed point on step



CABARET for uncompressible flows.

Stream instability on grid (256 x 256)

Flow behind turbulizing grid

Real stream CABARET simulation (256х512)

FLUENT simulation

Submerged jet on computational grid (128 x 640)

Foto

Result of simulation. Vorticity Field.

Animation

Towards Empiricism-Free Large Eddy Simulation for Thermo-Hydraulic Problems

Aanimation

15MC,Re=85000

Mixing hydrogen under containment

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

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.

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.

Thank you for your attention

Time reversed algorithm for pure convection V.М.Goloviznin.

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