Sixth Order Multiple Coarse Grid Computation for Solving 1D Partial Differential Equation

Abstract

We present a new method using multiple coarse grid computation technique to solve one dimensional (1D) partial
differential equation (PDE). Our method is based on a fourth order discretization scheme on two scale grids and
the Richardson extrapolation. For a particular implementation, we use multiple coarse grid computation to compute
the fourth order solutions on the fine grid and all the coarse grids. Since every fine grid point has a corresponding
coarse grid point with fourth order solution, the Richardson extrapolation procedure is applied for every fine grid
point to increase the order of solution accuracy from fourth order to sixth order. We compare the maximum absolute
error and the order of solution accuracy for our new method, the standard fourth order compact (FOC) scheme and
Wang-Zhang’s sixth order multiscale multigrid method. Two convection-diffusion problems are solved numerically
to validate our proposed method.