An Explicit Mapped Tent Pitching Scheme for Maxwell Equations

Abstract

An Explicit Mapped Tent Pitching Scheme forMaxwell EquationsJay Gopalakrishnan, Matthias Hochsteger, Joachim Sch ̈oberl, and ChristophWintersteigerAbstractWe present a new numerical method for solving time dependent Maxwellequations, which is also suitable for general linear hyperbolic equations. It is basedon an unstructured partitioning of the spacetime domain into tent-shaped regionsthat respect causality. Provided that an approximate solution is available at the tentbottom, the equation can be locally evolved up to the top of the tent. By mappingtents to a domain which is a tensor product of a spatial domain with a time in-terval, it is possible to construct a fully explicit scheme that advances the solutionthrough unstructured meshes. This work highlights a difficulty that arises when stan-dard explicit Runge Kutta schemes are used in this context and proposes an alter-native structure-aware Taylor time-stepping technique. Thus explicit methods areconstructed that allow variable time steps and local refinements without compro-mising high order accuracy in space and time. These Mapped Tent Pitching (MTP)schemes lead to highly parallel algorithms, which utilize modern computer archi-tectures extremely well.