An efficient energy conserving semi-Lagrangian kinetic scheme for the Vlasov-Ampère system

Abstract

In this paper, we present a novel kinetic scheme termed the Energy Conserving Semi-Lagrangian (ECSL) for the Vlasov-Ampère system. The novelty of the ECSL is that it retains the efficiency of the explicit scheme while simultaneously maintaining energy conservation and unconditional stability properties of the implicit scheme without relying on nonlinear iteration. The proposed ECSL method includes two main ingredients: the conservative Semi-Lagrangian (CSL) scheme and a novel field solver. The CSL scheme is utilized for the phase space discretization of the Vlasov equation, which enables the scheme to conserve mass exactly and remove the Courant-Friedrichs-Lewy restriction. The novel field solver is proposed by coupling the Ampère equation and the moments of the Vlasov equation in a semi-implicit way, allowing for an explicit and efficient calculation of the electric field. The combination of the novel field solver and CSL ensures that ECSL scheme conserves total energy and mass on the fully discrete level, regardless of spatial resolution and time step size. Moreover, the ECSL scheme still provides reliable solution even when its spatial and temporal resolution are insufficient to fully resolve the Debye length and plasma period, making it a promising tool for multiscale and lengthy simulations. Several numerical experiments are presented to demonstrate the accuracy, efficiency, and conservation properties of the proposed method.