Streamline methods for parabolic differential equations

Abstract

Parabolic advection-diffusion equations arise when modelling flow in porous media. We will in this thesis discuss two different problem set-ups from which these types of equations arise: Groundwater contamination with diffusion/dispersion; Fractional-flow formulation of immiscible two-phase flow. Streamline methods equipped with time-of-flight coordinates are attractive alternatives or supplements to traditional solution methods of advection-diffusion equations. This is particulary the case when cross-streamline diffusive effects can be neglected. In this case the possibly 3-dimensional equations can be reduced to 1-dimensional equations along the streamlines. If cross-streamline effects need to be taken into account, these effects can be simulated on background grids through mappings which introduce significant numerical diffusion. We propose a method to take care of the cross-streamline diffusive effects along normal-lines in 2D. It is based on operator splitting, reducing the 2D-equations to 1-dimensional equations along streamlines and normal-lines.