Time reversed algorithm for pure convection

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

2. Mathematical modeling transport phenomena on computational grid is one of the fundamental problems of the modern computational mathematics Simplest transport equation is time reversed is led to the same equation Substitution

3. Finite difference schemes of time reversed quality t x 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. One of them is well known Leap-Frog scheme Next one is Iserles scheme t x

4. Finite difference schemes of time reversed quality t x Implicit time reversible scheme is also well – known Sn Karlsons scheme Leap-Frog scheme is transformed into Arakawa – Lilly Schemes in multidimensional cases and successfully Explored in ocean modeling. Sn – Karlson scheme in the form of dSn-scheme is used In neutrons transport calculation for nuclear reactor. Explicit Iserles scheme is transformed into “CABARET” schemes, witch have a wide sphere of usability.

5. “CABARET” scheme Iserles scheme can be rewrite as t Variables will be called as “fluxes variables”. Variables will be noted as “conservative values” The next step of transform gives “two layers form” x

6. Dissipation and dispersive surfaces «Leap-Frog» «CABARET» Dispersion Dissipation

7. Since the CABARET scheme is second-order, according to the Godunov theorem it needs some procedure to enforcing monotonicity Maximum principle Consider 3 values inside 1 cell New principle item:direct application of maximum principle Adds on just enough dissipation needed for draining the energy from unresolved scales, “entropy” condition We constrain the solution so that

8. 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 n+1 n I I+1

9. 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 n+1 n I I+1

10. “Compact Accurately Boundary Adjusting high-REsolution Techniquefor Fluid Dynamics Computational stencil of the forerunner of CABARET scheme Another reason to call it CABARET…

11. CABARET for gas dynamics flows.First unique feature.

12. Gas dynamics:verification test Contact discontinuity Weak contact discontinuity Strong contact discontinuity independence from amplitude

13. First Unusual Feature of CABARET: independence shock wave thickness from amplitude Very slow shock wave Ordinary shock wave Very strong shock wave

14. Verification taskBlast Wave problem P,Woodward, P,Colella J,Comp,Phys,, 54, 115-173 (1984)‏

15. 1-D shock interaction with density perturbations: Shu&Osher problem Shock capturing capability without notable dissipation

16. 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 Grid (481x121) Grid (1921x481) Grid (961x241) Double Mach reflection test Titarev and Toro, 2002; J.Qiu and C.-W. Shu, 2003

17. CABARET for aeroacoustics problems.Second unique feature.

18. D2 acoustic Gaussian pulse propagation on nonuniform grid Second Unusual feature of CABARET: Acoustic disturbances is not dissipate Initial condition Computational grid:

19. D2 acoustic Gaussian pulse propagationon nonuniform grid

20. Simulation of vortex flow.Third unique feature.

21. 2-D zero-circulation compressible isentropic vortex in a periodic box Karabasov and Goloviznin, 2008 Full Euler equations are solved Stationary and stable solution to EE. But how long can the numerical scheme hold it? L=0.05 One revolution: T=1.047 H=1

22. Single Vortex Third Unusual feature of CABARET: Stationary vortex is not dissipate Presure Entropy Computational grid 50х50

23. Vortex Dipole

24. Conserves total k.e. within ~ 1% Vortex preserving capability: Problem of a steady 2-D zero-circulation compressible vortex in a periodic box domain Vorticity t=100 (30x30), 1.5 points per radius (p.p.r.) (120x120), 6 p.p.r. Vortex in a box: stationary and stable solution to the Euler equations. But how long can the numerical scheme preserve it? (60x60), 3 p.p.r.

25. Vortex preserving capability: what happens with a conventional 2nd-3rd order conservative method? (e.g., Roe-MUSCL-TVD, grid (240x240)) Vorticity (240x240) 12 points per vortex radius With the limiter the solution is too dissipative With the TVD limiter: t=4 With the TVD limiter: t=100 No limiter: t=4 Without the limiter it is too dispersive

26. 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 Zhou and Wei, 2003; Karabasov and Goloviznin, 2007 Weak vortex Strong vortex

27. D2 Backward Step Re=5000 40 greed point on step 10 greed point on step 20 greed point on step

28. CABARET for uncompressible flows.

29. Stream instability on grid (256 x 256)‏

30. Flow behind turbulizing grid CABARET simulation (256х512)‏ Real stream FLUENT simulation

31. Submerged jet on computational grid (128 x 640)‏ Foto Result of simulation. Vorticity Field. Animation

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

33. Aanimation 15MC,Re=85000

34. Mixing hydrogen under containment

35. 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

36. Conclusions Business problem: implementation of CABARET in the industry Innovative scientific problem: spreading of CABARET on new sphere of science and increasing order of accuracy up to fourth.

37. 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.: ИБРАЭРАН, 2005, -37 p.

38. Thank you for your attention