Semi-Lagrangian discontinuous Galerkin schemes for some first and second-order partial differential equations

TL;DR

This paper introduces explicit, unconditionally stable high-order semi-Lagrangian discontinuous Galerkin schemes for first- and second-order linear PDEs, with rigorous error estimates and practical tests demonstrating their effectiveness.

Contribution

It develops novel high-order, unconditionally stable semi-Lagrangian DG schemes for PDEs, including new error estimates and stability results for variable coefficients.

Findings

High-order schemes are unconditionally stable for linear PDEs.

Error estimates are sharp and applicable to variable coefficients.

Schemes perform well on academic test examples.

Abstract

Explicit, unconditionally stable, high-order schemes for the approximation of some first- andsecond-order linear, time-dependent partial differential equations (PDEs) are proposed.The schemes are based on a weak formulation of a semi-Lagrangian scheme using discontinuous Galerkin (DG) elements.It follows the ideas of the recent works of Crouseilles, Mehrenberger and Vecil (2010), Rossmanith and Seal (2011),for first-order equations, based on exact integration, quadrature rules, and splitting techniques for the treatment of two-dimensionalPDEs. For second-order PDEs the idea of the schemeis a blending between weak Taylor approximations and projection on a DG basis.New and sharp error estimates are obtained for the fully discrete schemes and for variable coefficients.In particular we obtain high-order schemes, unconditionally stable and convergent,in the case of linear first-order PDEs, or linear second-order PDEs with constant coefficients.In the case of non-constant coefficients, we construct, in some particular cases,"almost" unconditionally stable second-order schemes and give precise convergence results.The schemes are tested on several academic examples.