Operator splitting methods for systems of convection-diffusion equations: nonlinear error mechanisms and correction strategies

Abstract

Many numerical methods for systems of convection-diffusion equations are based upon an operator splitting formulation, where convective and diffusive forces are accounted for in separate substeps. We describe the nonlinear mechanism of the splitting error in such numerical methods in the onedimensional case, a mechanism that is intimately linked to the local linearizations introduced implicitly in the (hyperbolic) convection steps by the use of an entropy condition. For convection-dominated flows, we demonstrate that operator splitting methods typically generate a numerical widening of viscous fronts, unless the splitting step is of the same magnitude as the diffusion scale. To compensate for the potentially damaging splitting error, we propose a corrected operator splitting (COS) method for general systems of convection-diffusion equations with the ability of correctly resolving the nonlinear balance between the convective and diffusive forces. In particular, COS produces viscous shocks with correct structure also when the splitting step is large. A front tracking method for systems of conservation laws, which in turn relies heavily on a Riemann solver, constitutes an important part of our COS strategy. The proposed COS method is successfully applied to a system modelling two-phase, multicomponent flow in porous media and a triangular system modelling three-phase flow.