Convergence and performance of iterative methods for solving variable coefficient convection-diffusion equation with a fourth-order compact difference scheme

We conduct convergence analysis on some classical stationary iterative methods for solving the two-dimensional variable coefficient convection-diffusion equation discretized by a fourth-order compact difference scheme. Several conditions are formulated under which the coefficient matrix is guaranteed to be an M-matrix. We further investigate the effect of different orderings of the grid points on the performance of some stationary iterative methods, multigrid method, and preconditioned GMRES. Three sets of numerical experiments are conducted to study the convergence behaviors of these iterative methods under the influence of the flow directions, the orderings of the grid points, and the magnitude of the convection coefficients.