Time reversed algorithm for pure convection 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 Substitution is led to the same equation

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. x t One of them is well known Leap-Frog scheme Next one is Iserles scheme x t

Finite difference schemes of time reversed quality Implicit time reversible scheme is also well – known Sn Karlsons scheme x t 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.

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

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

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

Main distinguishes CABARET from 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+1 I 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 Weak contact discontinuity Strong contact discontinuity independence from amplitude

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

Verification task Blast Wave problem P,Woodward, P,Colella J,Comp,Phys,, 54, (1984)‏

1-D shock interaction with density perturbations: Shu&Osher problem 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 60 0 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 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=0.05One revolution: T=1.047 Stationary and stable solution to EE. But how long can the numerical scheme hold it? Full Euler equations are solved Karabasov and Goloviznin, 2008

Single Vortex Presure Entropy 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 2 nd -3 rd order conservative method? (e.g., Roe-MUSCL-TVD, grid (240x240)) With the TVD limiter: t=4 With the TVD limiter: t=100No 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= greed point on step 20 greed point on step 10 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 and increasing 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 – 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 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 Transport in Porous Media Volume 66, Number 3 / February, 2007 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 – )- М.: ИБРАЭ РАН, 2005, -37 p.

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