An efficient implicit boundary integral solver for the Vlasov-Maxwell system

Abstract

Summary form only given. Maxwell's equations are an integral part of many plasma models, and therefore robust, efficient solvers are needed in order to attain accurate numerical simulations of plasma. By formulating the Vlasov-Maxwell problem in terms of vector potentials, Maxwell's equation reduces to 4 wave equations, (one for the scalar potential and 3 for the vector potential), forced by the charges and currents as determined by the electron and ion distribution functions. Such problems are challenging because they often involve multiple time and spatial scales, complex geometries, and many particles, which are modeled as moving point sources. Explicit methods are efficient for spatial discretization, but can only take small time steps due to the CFL restriction, and are therefore are unfavorable for long time simulations. We present an implicit, A-stable boundary integral solver (IBIS) for the wave equation, using a method of lines transpose (MOLT) approach. We first build a fast O(N) solver in one spatial dimension, and then extend it to higher dimensions using alternate direction implicit (ADI) splitting, thus maintaining efficiency. This makes our IBIS as efficient as explicit solvers, but additionally allows for larger time steps and thus faster simulations. We can also address complex geometries, moving point sources, and a wide variety of boundary conditions. Additionally, we have implemented domain decomposition, which makes our algorithm flexible for use in adaptive mesh refinement.