Mapped tent pitching schemes for hyperbolic systems

TL;DR

This paper introduces a method to apply standard numerical discretizations within tent-shaped spacetime domains for hyperbolic systems, enabling flexible local time stepping by transforming equations into a separable domain.

Contribution

It presents a novel technique to map tent-shaped spacetime regions to separable domains, allowing the use of existing discretizations for hyperbolic systems in a flexible manner.

Findings

Successfully applied to acoustic wave equation

Extended to Euler system examples

Demonstrates compatibility with explicit time stepping

Abstract

A spacetime domain can be progressively meshed by tent shaped objects. Numerical methods for solving hyperbolic systems using such tent meshes to advance in time have been proposed previously. Such schemes have the ability to advance in time by different amounts at different spatial locations. This paper explores a technique by which standard discretizations, including explicit time stepping, can be used within tent-shaped spacetime domains. The technique transforms the equations within a spacetime tent to a domain where space and time are separable. After detailing techniques based on this mapping, several examples including the acoustic wave equation and the Euler system are considered.