A Fast Cascadic Multigrid Method for Exponential Compact FD Discretization of Singularly Perturbed Convection-Diffusion Equations

Abstract

Solving singularly perturbed convection-diffusion equations, especially for 3D problems, is a challenging problem. In this paper, we extend our work on the extrapolation cascadic multigrid (EXCMG) method for solving the 3D Poisson equation — cf. [Pan et al., J. Sci. Comput. 2017], to 3D convection-diffusion equations with singularly perturbed parameters. First, we introduce an exponential higher order compact finite difference scheme to discretize the 3D convection-diffusion equation with variable convection coefficients, resulting in a larger-scale nonsymmetric linear system. Then, we propose an EXCMG method combined with the biconjugate gradient stabilized smoother to solve the larger-scale nonsymmetric system efficiently. Numerical experiments demonstrate that the EXCMG method is a highly effective solver for convection-dominated problems, which outperforms the existing multigrid methods such as aggregation-based algebraic multigrid method.