A Characteristic Finite Element Method for Solving Non-Linear Convection-Diffusion Equations on Locally Refined Grids

A method for solving nonlinear convection-diffusion equations in two or three space dimensions is described. These equations play an important role in the numerical simulation of immiscible, two-phase flow through porous media. All computations are performed on locally refined and dynamically adapted grids. This increases efficiency and ensures an optimal representation of shock fronts. Operator-splitting is used to decouple convection and diffusion, which reduces the problem to an alternating sequence of hyperbolic and elliptic equations. An accurate characteristic method deals with the hyperbolic equations. Nonlinearities in the convection term are treated by solving Riemann problems along stream lines. Elliptic equations are discretised by mixed finite elements and solved by multi-grid. Gravity effects are included by a spatial splitting of the convection term. The method induces almost no numerical diffusion. It also permits to use large time steps and it conserves mass exactly. Numerical results are presented which demonstrate the performance of the method for some multi-dimensional test problems.