Stability of structure-aware Taylor methods for tents

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 s -stage SAT timestepping within a tent is weakly stable under the time step constraint Δ t ≤ C h 1 + 1 / s \Delta t \leq Ch^{1+1/s} , where Δ t \Delta t is the time step size and h 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.