Block iterative methods for the nine-point approximation to the convection-diffusion equation

We study the numerical solution of a steady convection-diffusion equation in which the convection term is dominant. A nine-point fourth order difference scheme is examined and the block successive overrelaxation (BSOR) and the block alternating group explicit (BLAGE) iterative methods are considered for solving the sparse unsymmetric linear system. When the diffusion coefficient is small, the eigenvalues of the block Jacobi iteration matrix almost lie on the imaginary axis and using bounds for their eigenspectrum, we show that the linear system can be effectively solved by the block Gauss-Seidel method. The size of the linear system is reduced by the Strides of 3 algorithm and the BSOR and BLAGE methods are applied to the reduced system of linear equations. This leads to greater convergence for both methods. In addition, a great deal of parallelism can be achieved by using BLAGE.