A second-order structure- and positivity-preserving convex limiting method for the Vlasov equations

Abstract

In this paper, we introduce a novel second-order, positivity-preserving finite element method for the Vlasov equations using a convex limiting algorithm. The method employs strong-stability-preserving (SSP) Runge-Kutta time integration and a tensor-product construction of the phase-space mesh for efficient high-dimensional implementations. The convex limiting algorithm combines the robust first-order positivity-preserving graph viscosity approach with high-order residual-based viscosity stabilization to obtain a high-order positivity-preserving scheme. Both novel first-order and high-order methods applicable to high-dimensional problems such as the Vlasov system are presented. In addition, we propose a divergence-cleaning technique for Maxwell's equations to ensure that the divergence constraints of the electromagnetic fields are satisfied. Numerical experiments are provided to demonstrate the accuracy and robustness of the proposed methods.