Hermite Spline Interpolation on Patches for Parallelly Solving the Vlasov-Poisson Equation

Abstract

This work is devoted to the numerical simulation of the Vlasov equation using a phase space grid. In contrast to ParticleIn-Cell (PIC) methods, which are known to be noisy, we propose a semi-Lagrangian-type method to discretize the Vlasov equation in the two-dimensional phase space. As this kind of method requires a huge computational effort, one has to carry out the simulations on parallel machines. For this purpose, we present a method using patches decomposing the phase domain, each patch being devoted to a processor. Some Hermite boundary conditions allow for the reconstruction of a good approximation of the global solution. Several numerical results demonstrate the accuracy and the good scalability of the method with up to 64 processors. This work is a part of the CALVI project.

Comparison of Vlasov solvers for spacecraft charging simulation

“…If we treat them separately on different processors, we have to take care of interface conditions. A parallelization procedure of the semi-Lagrangian method is presented in [13]. (3) The Back-Trajectory method becomes particularly interesting when we use the geostationary approximation.…”

“…Contrary to the case of periodic boundary conditions (see [13]), the values of the function outside [x 0 , x N ] can not be known. Therefore we have to use the values of f at time t n inside this interval to compute the distribution function at time t n + Δt.…”

“…We refer e.g. to [13,30] for details on this interpolation step, an alternative approach based on the WENO procedure has been proposed recently in [3]. However, most of the references we are aware of restrict to periodic boundary conditions and do not address the question of the interpolation rule to be adopted for boundary points.…”

“…To this end, we have to solve (2.2). This resolution is splitted into three steps [1,13,30]: -backward advection of Δt/2 in the spatial direction:…”

A theoretical and simulation study of the contact discontinuities based on a Vlasov simulation code

“…Vlasov simulations based on the semi-Lagrangian approach by using the cubic spline interpolation operator have been carried out by Cheng and Knorr [1976] and by Crouseilles et al [2007, 2009, andreferences therein]. Crouseilles et al [2007Crouseilles et al [ , 2009 also showed how to use the cubic spline to determine the derivatives of a tabulate function with equal mesh size for a given Hermite boundary conditions. In Appendix B, we demonstrate that we can use the cubic spline to determine the derivatives of a non-equal-spacing tabulated function with either periodic or nonperiodic boundary conditions.…”