A Charge Preserving Scheme for the Numerical Resolution of the Vlasov-Ampère Equations

Abstract

In this report, a charge preserving numerical resolution of the 1D Vlasov-Ampère equation is achieved, with a forward Semi-Lagrangian method introduced in [10]. The Vlasov equation belongs to the kinetic way of simulating plasmas evolution, and is coupled with the Poisson's equation, or equivalently under charge conservation, the Ampère's one, which self-consistently rules the electric field evolution. In order to ensure having proper physical solutions, it is necessary that the scheme preserves charge numerically. B-Spline deposition will be used for the interpolation step. The solving of the characteristics will be made with a Runge-Kutta 2 method and with a Cauchy-Kovalevsky procedure.

Hamiltonian splitting for the Vlasov–Maxwell equations

“…First, a generalization to finite volumes methods for the spatial discretization have to be performed, taking care of the charge conservation property. We are confident this can be achieved by combining ideas of [24] and [11]. Second, a more detailed study of high order splittings (such as the triple Jump scheme) is necessary, since the use of small steps due to the CFL condition imposed by the Maxwell equations is a limiting factor.…”

“…Finally, let us emphasize that it is well known that for both approaches (PIC and deterministic), the Poisson equation needs to be satisfied to machine precision (even if Poisson's equation is not solved directly in the numerical method). This is the charge conservation problem ( [24,5,17,25,1,11]).…”

Numerical simulations of one laser-plasma model based on Poisson structure

“…In comparison, gridbased methods offer a good alternative to overcome this lack of precision, for instance the semi-Lagrangian methods [18,13,35], energy-conserving discontinuous Galerkin methods [10,11], and so on. As for the LP model and its variants, there have been some research works [3,5,6,8,14,20,22,24,2,3,21,36], in which mathematical analyses of the LP model, some numerical methods and one convergence analysis have been studied.…”

Numerical study of the two-species Vlasov–Ampère system: Energy-conserving schemes and the current-driven ion-acoustic instability

“…While there are abundant literature on grid-based Vlasov solver for single-species VA or Vlasov-Poisson (VP) system, e.g. [7,34,4,23,26,15,21], there are relatively fewer published works for the two-species system. In [19,20,18], Fourier transformed methods are used to compute two-species VP system for electron and ion holes.…”