Analysis of stationary iterative methods for the discrete convection–diffusion equation with a 9-point compact scheme

We study the convergence of point and line stationary iterative methods for solving the linear system arising from a fourth-order 9-point compact finite difference discretization of the two-dimensional convection–diffusion equation with constant coefficients. We present new techniques to bound the spectral radii of iteration matrices in terms of the cell Reynolds numbers. We also derive analytic formulas for the spectral radii for special values of the cell Reynolds numbers and study asymptotic behaviors of the analytic bounds. The results provide rigorous justification for the numerical experiments conducted elsewhere, which show good stability for the fourth-order compact scheme. In addition, we compare the 9-point scheme with the traditional 5-point difference discretization schemes and conduct some numerical experiments to supplement our analyses.