Stability of structure-aware Taylor methods for tents

TL;DR

This paper analyzes the stability of structure-aware Taylor methods for tent-shaped spacetime regions, showing weak stability under specific timestep constraints and providing numerical verification of these theoretical results.

Contribution

It introduces stability analysis for SAT methods within tents, establishing weak stability conditions and demonstrating improved stability with high-order discretizations.

Findings

Weak stability under timestep constraint t h^{1+1/s}

Enhanced stability with high-order SAT and low-order spatial polynomials

Numerical verification confirms theoretical stability estimates

Abstract

Structure-aware Taylor (SAT) methods are a class of timestepping schemes designed for propagating linear hyperbolic solutions within a tent-shaped spacetime region. Tents are useful to design explicit time marching schemes on unstructured advancing fronts with built-in locally variable timestepping for arbitrary spatial and temporal discretization orders. The main result of this paper is that an $s$-stage SAT timestepping within a tent is weakly stable under the time step constraint $\Delta t \leq Ch^{1+1/s}$, where $\Delta t$ is the time step size and $h$ is the spatial mesh size. Improved stability properties are also presented for high order SAT time discretizations coupled with low order spatial polynomials. A numerical verification of the sharpness of proven estimates is also included.