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

We present an explicit fourth-order compact nite diierence scheme for approximating the three dimensional convection-diiusion equation with variable coeecients. This 19-point formula is deened on a uniform cubic grid. Fourier smoothing analysis is performed to show that the smoothing factor of certain relaxation techniques used with the scheme is smaller than 1. We design a parallelization-oriented multigrid method for fast solution of the resulting linear system using a four-color Gauss-Seidel relaxation technique for robustness and eeciency, and a scaled residual injection operator to reduce the cost of multigrid inter-grid transfer operator. Numerical experiments on a 16 processor vector computer are used to test the high accuracy of the discretization scheme as well as the fast convergence and the parallelization or vectorization eeciency of the solution method. Several test problems are solved and highly accurate solutions of the 3D convection-diiusion equations are obtained for small to medium values of the grid Reynolds number. EEects of using diierent residual projection operators are compared on both vector and serial computers.