On Semi‐Lagrangian Exponential Integrators and Discontinuous Galerkin Methods

Discontinuous Galerkin (dG) finite‐element methods are well‐known to be suitable for solving convection‐dominated convection diffusion problems. High order Runge‐Kutta methods such as the RKDG and SSP methods have been developed and tested to work quite well (see e.g., Cockburn & Shu (2001), Gottlieb et al. (2001,2009)). An issue of concern remains the strong CFL restrictions on the discretization parameters. Restelli et al. (2006) proposed combining the dG methods with the semi‐Lagrangian methods in what they called semi‐Lagragian discontinuous Galerkin (SLDG). Proposed implementations of the SLDG methods however suffer from limited spatial accuracy as opposed to the RKDG methods. We hereby propose a method in the framework of Lie‐group exponential integrators (see Cellodoni et al. [1, 2]), that uses a modified version of the SLDG method as a building block for computing compositions of convection flows, and establish high temporal accuracy while maintaining the good properties of the dG formulations.