High accuracy multigrid solution of the 3D convection-diffusion equation

We present an explicit fourth-order compact finite difference scheme for ap proximating the three-dimensional (3D) convection-diffusion equation with v ariable coefficients. This 19-point formula is defined on a uniform cubic g rid. Fourier smoothing analysis is performed to show that the smoothing fac tor of certain relaxation techniques used with the scheme is smaller than 1 . We design a parallelization-oriented multigrid method for fast solution o f the resulting linear system using a four-color Gauss-Seidel relaxation te chnique for robustness and efficiency, and a scaled residual injection oper ator to reduce the cost of multigrid inter-grid transfer operator. Numerica l experiments on a 16 processor vector computer are used to test the high a ccuracy of the discretization scheme as well as the fast convergence and th e parallelization or vectorization efficiency of the solution method. Sever al test problems are solved and highly accurate solutions of the 3D convect ion diffusion equations are obtained for small to medium values of the grid Reynolds number. Effects of using different residual projection operators are compared on both vector and serial computers. (C) 2000 Elsevier Science Inc. All rights reserved.