Improved computation of non-linear advection in porous media using slightly modified basic finite elements algorithms

The numerical stability of standard finite element schemes applied to the advection-diffusion equation is evaluated using a space-time eigenvalue analysis. Unlike the usual approaches which only consider temporal aspects of stability, this analysis also describes the spatial stability of the solutions. To this end, the one-dimensional advection-diffusion equation is put into an alternative semi-discrete form which allows the derivation of a very practical stability condition. In multidimensional flow situations the latter is applied along the streamlines by means of a tensorial corrective function that prevents excessive numerical smearing of fronts or phase interfaces. The efficiency of the procedure is illustrated by an example which successfully simulates the coupling of two low miscible fluid phases in a variably saturated porous medium.