Solving the advection-diffusion equation on unstructured meshes with discontinuous/mixed finite elements and a local time stepping procedure

Explicit schemes are known to provide less numerical diffusion in solving the advection-diffusion equation, especially for advection-dominated problems. Traditional explicit schemes use fixed time steps restricted by the global CFL condition in order to guarantee stability. This is known to slow down the computation especially for heterogeneous domains and/or unstructured meshes. To avoid this problem, local time stepping procedures where the time step is allowed to vary spatially in order to satisfy a local CFL condition have been developed. In this paper, a local time stepping approach is used with a numerical model based on discontinuous Galerkin/mixed finite element methods to solve the advection-diffusion equation. The developments are detailed for general unstructured triangular meshes. Numerical experiments are performed to show the efficiency of the numerical model for the simulation of (i) the transport of a solute on highly unstructured meshes and (ii) density-driven flow, where the velocity field changes at each time step. The model gives stable results with significant reduction of the computational cost especially for the non-linear problem. Moreover, numerical diffusion is also reduced for highly advective problems. Copyright (C) 2009 John Wiley & Sons, Ltd.