For a given initial condition at the bottom of a tent, the discrete equations may be solved within each individual tent, up to the tent top. Discontinuous Galerkin (DG) methods are of particular interest since they offer a systematic avenue to build high order methods. A tent pitching method requires a numerical scheme to discretize the equation on that mesh. It is not limited to Maxwell equations, but can be applied to general hyperbolic equations. A tent pitching method introduces a special “causal” spacetime mesh that respects this finite speed of propagation. Thus, the field at a certain point in space and time depends only on field values within a dependency cone. These Mapped Tent Pitching (MTP) schemes lead to highly parallel algorithms, which utilize modern computer architectures extremely well.Įlectromagnetic waves propagate at the speed of light. Thus explicit methods are constructed that allow variable time steps and local refinements without compromising high order accuracy in space and time. 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. 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. Provided that an approximate solution is available at the tent bottom, the equation can be locally evolved up to the top of the tent.
It is based on an unstructured partitioning of the spacetime domain into tent-shaped regions that respect causality. We present a new numerical method for solving time dependent Maxwell equations, which is also suitable for general linear hyperbolic equations.
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