  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.
