A Discretisation for Transport Problems with Dominant Convection Using Characteristics and Finite Elements

Summary

In this paper a discretisation for two-dimensional steady state transport processes in porous media is presented. To guarantee stability and good accuracy of the discrete solution even in the convection dominated case two mean concepts are used: The convective part of the equation is discretised by using the method of characteristics. Mixed finite element / finite volume techniques are applied to the resulting modified equation. The discretisation can be applied also to the hyperbolic limit problem of transport without dispersion. Some examples are presented. They demonstrate the stability of the discretisation and the sensitivity with respect to the choice of the steplength parameter for the discrete characteristics. For convection dominated problems the linear systems achieved by this discretisation are singular perturbations of subdiagonal systems, if a ‘flow oriented’ numbering of the finite element nodes is used. Gauss-Seidel and SOR-algorithm with overrelaxation can be used as efficient solvers.