Integrated fast and high-accuracy computation of convection diffusion equations using multiscale multigrid method

We present an explicit sixth-order compact finite difference scheme for fast high-accuracy numerical solutions of the two-dimensional convection diffusion equation with variable coefficients. The sixth-order scheme is based on the well-known fourth-order compact (FOC) scheme, the Richardson extrapolation technique, and an operator interpolation scheme. For a particular implementation, we use multiscale multigrid method to compute the fourth-order solutions on both the coarse grid and the fine grid. Then, an operator interpolation scheme combined with the Richardson extrapolation technique is used to compute a sixth-order accurate fine grid solution. We compare the computed accuracy and the implementation cost of the new scheme with the standard nine-point FOC scheme and Sun-Zhang's sixth-order method. Two convection diffusion problems are solved numerically to validate our proposed sixth-order scheme.