L2-stability analysis of IMEX-(σ,μ)DG schemes for linear advection-diffusion equations

Abstract

A fully discrete L2-stability analysis is carried out for linear advection-diffusion problems in one space dimension, discretized in space by discontinuous Galerkin methods based on the (σ,μ)-family of diffusion schemes and implicit-explicit Runge-Kutta schemes in time. Advection terms are discretized explicitly while diffusion terms are solved implicitly. Conditions on the parameters σ and μ are derived which guarantee L2-stability for time steps Δt=O(d/a2), where a and d denote the advection and diffusion coefficient, respectively, i.e. the allowable time step size does not decrease under grid refinement. These conditions are fulfilled in particular by the BR2 scheme as well as the recent (14,94)-recovery scheme. However, the BR1 scheme and the Baumann-Oden method do not possess a similar stability property which is also confirmed by numerical experiments.