An asymptotic preserving scheme for the Vlasov-Poisson-Fokker-Planck system in the high field regime. (English) Zbl 1265.82006
Summary: The Vlasov-Poisson-Fokker-Planck system under the high field scaling describes the Brownian motion of a large system of particles in a surrounding bath, where both collision and field effects (electrical or gravitational) are dominant. The numerical solution of this system is challenging due to the stiff collision term and stiff nonlinear transport term with respect to the high field. We present a class of asymptotics-preserving scheme which is efficient in the high field regime, namely, large time steps and coarse meshes can be used, yet the high field limit is still captured. The idea is to combine the two stiff terms and treat them implicitly. Thanks to the linearity of the collision term, using the discretization described in the paper by S. Jin and B. Yan [J. Comput. Phys. 230, No. 17, 6420–6437 (2011; Zbl 1408.76594)], we only need to invert a symmetric matrix. This method can be easily extended to higher dimensions. The method is shown to be positive, stable, mass and asymptotics preserving. Numerical experiments validate its efficiency in both kinetic and high field regimes including mixing regimes.
MSC:
82D10 | Statistical mechanics of plasmas |
82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
82C80 | Numerical methods of time-dependent statistical mechanics (MSC2010) |