On the Solution of the Convection-Diffusion Equation by Iteration

The discretization of the two-dimensional convection-diffusion equation usually leads to a linear system whose matrix coefficient is block two-cyclic consistently ordered. For the solution of the resulting linear system, several efficient stationary iterative methods were proposed, among others, by Chin and Manteuffel (1988), Elman and Golub (1990), de Pillis (1991) and Eiermann, Niethammer and Varga (1992). In the present work, we propose, as an alternative, the stationary Modified Successive Overrelaxation (MSOR) method or an "equivalent" 2-step method applied to the cyclically reduced linear system. It is shown both theoretically and experimentally that the application of a "continuous" version of Manteuffel's algorithm to derive the optimal parameters produces an iterative method that is asymptotically faster than the previous methods.