Parallelizing the Kolmogorov Fokker Planck Equation

Gerardo-Giorda, Luca; Tran, Minh-Binh

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 2, pp. 395-420.

We design two parallel schemes, based on Schwarz Waveform Relaxation (SWR) procedures, for the numerical solution of the Kolmogorov equation. The latter is a simplified version of the Fokker–Planck equation describing the time evolution of the probability density of the velocity of a particle. SWR procedures decompose the spatio-temporal computational domain into subdomains and solve (in parallel) subproblems, that are coupled through suitable conditions at the interfaces to recover the solution of the global problem. We consider coupling conditions of both Dirichlet (Classical SWR) and Robin (Optimized SWR) types. We prove well-posedeness of the schemes subproblems and convergence for the proposed algorithms. We corroborate our findings with some numerical tests.

Classification : 35K55, 65M12, 65M55
Mots clés : Domain decomposition, Schwarz waveform relaxation methods, optimized Schwarz, Kolmogorov equation, Fokker–Plank equation, kinetic equations

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