Theoretical and Practical Aspects of Space-Time DG-SEM Implementations

TL;DR

This paper compares two formulations of space-time DG-SEM, analyzing their theoretical equivalence, practical differences, and implementation challenges, supported by numerical experiments validating their accuracy for high-dimensional problems.

Contribution

It provides a detailed comparison of space-time DG-SEM approaches, highlighting their practical differences, implementation considerations, and validating their effectiveness through numerical experiments.

Findings

Both approaches are mathematically equivalent in solutions.

Practical differences affect software structure and solver interaction.

Numerical results confirm accuracy for 4D problems.

Abstract

We discuss two approaches for the formulation and implementation of space-time discontinuous Galerkin spectral element methods (DG-SEM). In one, time is treated as an additional coordinate direction and a Galerkin procedure is applied to the entire problem. In the other, the method of lines is used with DG-SEM in space and the fully implicit Runge-Kutta method Lobatto IIIC in time. The two approaches are mathematically equivalent in the sense that they lead to the same discrete solution. However, in practice they differ in several important respects, including the terminology used to describe them, the structure of the resulting software, and the interaction with nonlinear solvers. Challenges and merits of the two approaches are discussed with the goal of providing the practitioner with sufficient consideration to choose which path to follow. Additionally, implementations of the two methods are provided as a starting point for further development. Numerical experiments validate the theoretical accuracy of these codes and demonstrate their utility, even for 4D problems.