An Active Flux Method for Maxwell's equations

Abstract

We present a novel Active Flux method and a divergence-free method using exact evolution operators for solving Maxwell's equations in three space dimensions on Cartesian grids. The proposed methods are fully discrete, third-order accurate and rely on a truly multidimensional evolution operator based on Kirchhoff's formula for the electromagnetic wave equation. The first method is a classical Active Flux method with globally continuous reconstruction while the second method is an adaptation of the Active Flux method to a locally divergence-free reconstruction by Balsara. The applied reconstruction is built from facial degrees of freedom and is no longer globally continuous. The classical method keeps the divergence error bounded. The divergence-free method with face averages initialized by a potential keeps the constraints at round-off. We compare the accuracy of both methods for smooth and discontinuous test problems.