Robust Multigrid Methods for Convection Diffusion Problems with Closed Characteristics

Klaus Johannsen

Institute for Computer Applications 3
University of Stuttgart
Allmandring 5b
70569 Stuttgart
Germany


Abstract

We consider the two-dimensional convection diffusion equation with closed characteristics on unstructured triangular meshes and study problems with dominant convection. We present a detailed analysis for a representative number of model problems. The investigation is based on Fourier analysis in the case of a very simple model problem and on numerical experiments for more realistic situations. The experiments have been carried out on the software platform UG using grids with up to 400.000 unknowns to investigate the asymptotic behaviour. As a result we present a multigrid scheme for this class of problems which is robust w.r.t. to the Peclet number as well as to the amount of crosswind diffusion introduced by the discretization scheme. The method is based on a robust Gauss-Seidel type smoother using an ordering strategy and a special treatment of the cyclic dependencies. Moreover a modification of the coarse grid correction is neccessary to improve the discrete approximation property. The presented method is of optimal complexity (up to a logarithmic factor).

For practical use (problems with non-vanishing crosswind diffusion on grids up to 1.000.000 unknowns) we show that a modification based on Krylov space methods leads to an robust method of optimal complexity. The resulting algorithm is well suited for more complex problems in which dominant convection plays an important role. An application to the density driven groundwater flow is discussed.