Stochastic approach to numerical solution of Fokker--Planck equations

M. F. Ivanov ; V. A. Gal'burt

Matematičeskoe modelirovanie, Tome 20 (2008) no. 11, pp. 3-27

Résumé

It is considered the generalization of a method of particles in cells for numerical modeling of plasma dynamics in view of influence of coulomb collisions, and also for the solution of the equations of convective transport of impurity in environments with complex structure of turbulent flows. The approach is based on the solution of the Fokker–Planck equation describing as a kinetic of collisional plasmas, so tasks of convective diffusion by replacement of the equation of diffusion type by equivalent system of the Langevin stochastic differential equations (SDE), which realize stochastic process of Markov type and are formally considered in this case as the equations of movement for modeling microparticles. It is established the connection between coefficients of the Fokker–Planck equation and Langevin equations, ensuring stochastic equivalence of these equations in result of which the function of distribution of realizations of stochastic process satisfies to the kinetic equation the Fokker–Planck. The algorithm of the numerical decision SDE is given. The concrete expressions for coefficients SDE, appropriate to the Landau kinetic equation for collisional plasma, to the kinetic equation in Lorentz approach and equation of turbulent diffusion are submitted. The role of collisions is established at acceleration of heavy ions in expanding plasma and during absorption of laser radiation by plasma of overcritical density.