DeepAI AI Chat
Log In Sign Up

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

by   Lea M. Versbach, et al.

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.


page 21

page 23

page 25

page 27


A space-time Trefftz discontinuous Galerkin method for the linear Schrödinger equation

A space-time Trefftz discontinuous Galerkin method for the Schrödinger e...

A Stable FE Method For the Space-Time Solution of the Cahn-Hilliard Equation

In its application to the modeling of a mineral separation process, we p...

A space-time hybridizable discontinuous Galerkin method for linear free-surface waves

We present and analyze a novel space-time hybridizable discontinuous Gal...

An experimental comparison of a space-time multigrid method with PFASST for a reaction-diffusion problem

We consider two parallel-in-time approaches applied to a (reaction) diff...