On Splitting-Based Numerical Methods for Convection-Diffusion Equations

Convection-diffusion equations model a variety of physical phenomena. Computing solutions of these equations is an important and challenging problem, especially in the convection dominated case, in which viscous layers are so thin that one is forced to use underresolved methods that may be unstable. If an insufficient amount of physical diffusion is compensated by an excessive numerical viscosity, the underresolved method is typically stable, but the resolution may be severely affected. At the same time, the use of dispersive schemes may cause spurious oscillations that may, in turn, trigger numerical instabilities. In this paper, we review a special operator splitting technique that may help to overcome these difficulties by numerically preserving a delicate balance between the convection and diffusion terms, which is absolutely necessary when an underresolved method is used. We illustrate the performance of the splitting-based methods on a number of numerical examples including the polymer system arising in modeling of the flooding processes in enhanced oil recovery, systems modeling the propagation of a passive pollutant in shallow water, and the incompressible Navier-Stokes equations.