Discontinuous Galerkin time integration for second-order differential problems: formulations, analysis, and analogies

TL;DR

This paper explores Discontinuous Galerkin methods as time integrators for second-order oscillatory systems, providing new convergence analyses, establishing equivalences with classical schemes, and comparing accuracy and efficiency.

Contribution

It introduces new convergence results for DG discretizations of second-order problems and proves their equivalence to classical time-stepping methods.

Findings

New convergence analyses for second-order DG schemes

Proofs of equivalence between DG and classical methods

Comparative assessment of accuracy and computational cost

Abstract

We thoroughly investigate Discontinuous Galerkin (DG) discretizations as time integrators for second-order oscillatory systems, considering both second-order and first-order formulations of the original problem. Key contributions include new convergence analyses for the second-order formulation and equivalence proofs between DG and classical time-stepping schemes (such as Newmark schemes and general linear methods). In addition, the chapter provides a detailed review and convergence analysis for the first-order formulation, alongside comparisons of the proposed schemes in terms of accuracy, consistency, and computational cost.