JOURNAL OF COMPUTATIONAL PHYSICS 132, 123–129 (1997) ARTICLE NO. CP965627 A Compact Multigrid Solver for Convection-Diffusion Equations Murli M. Gupta,* Jules Kouatchou, and Jun Zhang Department of Mathematics, The George Washington University, Washington, DC 20052 E-mail: *mmg@math.gwu.edu Received April 29, 1996; revised October 30, 1996 In Section 2 we present the NPF difference scheme to solve the convection-diffusion equation (1), describe its We introduce a multigrid algorithm to solve the convection- diffusion equations using a nine-point compact difference scheme. implementation with multigrid, and carry out a Fourier We test the efficiency of the algorithm with various smoothers and smoothing analysis of the Gauss–Seidel operator. In Sec- intergrid transfer operators. The algorithm displays a grid-indepen- tion 3 we present numerical experiments that demonstrate dent convergence rate and produces solutions with high accuracy. the effectiveness and accuracy of the multigrid algorithm. Numerical results are presented to validate the conclusions.  1997 The paper ends with discussion and conclusions. Academic Press 2. THE MULTIGRID IMPLEMENTATION 1. INTRODUCTION 2.1. The Compact Nine-Point Stencil Consider the two-dimensional convection-diffusion Let h(=1/ N ) be the uniform mesh-size. The finite differ- equation ence approximation for convection-diffusion equation (1) at a grid point (x, y) which is denoted by x 0 (Fig. 1) involves Lu  u xx + u yy + u x +  u y = f (x, y), (x, y)  , (1) eight neighboring mesh points at (x  h, y), (x, y  h), u(x, y) = g (x, y), (x, y)  , (x  h, y  h). These points are denoted by x i , i = 1, 2, ..., 8, and the values of a function u at the point i are denoted which often appears in the description of transport phe- by u i . A compact fourth order approximation (NPF ) of nomena. The magnitudes of  and  determine ratios of the convection-diffusion equation (1) is given by [6] convection to diffusion. In many problems of practical interest the convective terms dominate the diffusion and the values of  and  are large. Many numerical simulations  8 i=0  i u i = h 2 2 [( f 4 + f 3 + f 2 + f 1 + 8f 0 ) (2) of (1) become increasingly difficult (converge slowly or even diverge) as the ratio of convection to diffusion in- + ( f 1 - f 3 ) +  ( f 2 - f 4 )], creases. For convection-diffusion equations with constant coef- where u i and  i are depicted by the stencil in Fig. 1 and ficients (1), Gupta et al. [6] proposed a compact fourth-  = h/2 and  = h/2 are the cell Reynolds numbers. order finite difference scheme which was shown to be both When the cell Reynolds numbers  and  are greater accurate and cost-effective; it is also stable for all values than 1.0, the coefficient matrix is no longer an M-matrix of  and . In [7], this compact nine-point formula (NPF ) (see [13] for the definition of M-matrix). The numerical was extended to solve convection-diffusion equations with experiments conducted in [6] showed that this scheme con- variable coefficients. The new scheme also has a truncation verges for any values of  and  when classical iterative error of order h 4 and the resulting systems of linear equa- methods are used. tions could be solved by classical iterative methods for large values of  and . 2.2. The Multigrid Method In this paper, we present a method that combines multigrid techniques with NPF to solve the convection- The multigrid method is among the fastest and most efficient iterative algorithms for solving linear systems aris- diffusion equation (1). For a wide range of  and , we compare the effectiveness of a number of smoothers for ing from discretizing elliptic differential equations (see [3, 12]). This method offers convergence rates independent solving the linear systems resulting from the use of NPF . We also compare the choices of the intergrid transfer oper- of the grid size and is very effective for solving large scale computation-intensive problems. Structurally, the algo- ators. 123 0021-9991/97 $25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.