# An Explicit Mapped Tent Pitching Scheme for Maxwell Equations

## TL;DR

This paper introduces a novel explicit numerical scheme for Maxwell equations using unstructured spacetime tents, enabling high-order accuracy, variable time steps, and efficient parallel computation on modern architectures.

## Contribution

It develops a new mapped tent pitching method that overcomes limitations of standard explicit schemes for hyperbolic PDEs, allowing flexible, high-order, and parallelizable solutions.

## Key findings

- Enables explicit high-order solutions on unstructured meshes.
- Supports variable time steps and local mesh refinement.
- Achieves efficient parallelization suitable for modern architectures.

## Abstract

We present a new numerical method for solving time dependent Maxwell equations, which is also suitable for general linear hyperbolic equations. It is based on an unstructured partitioning of the spacetime domain into tent-shaped regions that respect causality. Provided that an approximate solution is available at the tent bottom, the equation can be locally evolved up to the top of the tent. By mapping tents to a domain which is a tensor product of a spatial domain with a time interval, it is possible to construct a fully explicit scheme that advances the solution through unstructured meshes. This work highlights a difficulty that arises when standard explicit Runge Kutta schemes are used in this context and proposes an alternative structure-aware Taylor time-stepping technique. Thus explicit methods are constructed that allow variable time steps and local refinements without compromising high order accuracy in space and time. These Mapped Tent Pitching (MTP) schemes lead to highly parallel algorithms, which utilize modern computer architectures extremely well.