Abstract
In this paper, we propose an extrapolation full multigrid (EXFMG) algorithm to solve the large linear system arising from a fourth-order compact difference discretization of two-dimensional (2D) convection diffusion equations. A bi-quartic Lagrange interpolation for the solution on previous coarser grid is used to construct a good initial guess on the next finer grid for V- or W-cycles multigrid solver, which greatly reduces the number of relaxation sweeps. Instead of performing a fixed number of multigrid cycles as used in classical full multigrid methods, a series of grid level dependent relative residual tolerances is introduced to control the number of the multigrid cycles. Once the fourth-order accurate numerical solutions are obtained, a simple method based on the midpoint extrapolation is employed for the fourth-order difference solutions on two-level grids to construct a sixth-order accurate solution on the entire fine grid cheaply and directly. Numerical experiments are conducted to verify that the proposed method has much better efficiency compared to classical multigrid methods. The proposed EXFMG method can also be extended to solve other kinds of partial differential equations.
Highlights
Richardson extrapolation [1, 2] is an acceleration method, used to improve the rate of convergence for a sequence
We extend the idea described in the literature [32, 33, 39] to the original full multigrid method, and develop an extrapolation full multigrid (EXFMG) method to solve the 2D convection–diffusion equation with a fourth-order compact difference discretization
Taking the above four extrapolation formulas (5), (6), (7) and (8), we can obtain an extrapolation solution Eu, which is more accurate than the fourth-order accurate numerical solution uh/2 on the fine grid h/2. We denote this procedure from uh, uh/2 to Eu as an operator, we call it as fine mesh Richardson extrapolation (FMRE) interpolator
Summary
Richardson extrapolation [1, 2] is an acceleration method, used to improve the rate of convergence for a sequence. In 2009, Wang and Zhang designed an explicit sixth-order compact discretization strategy (MGSix) for the 2D Poisson equation [39] They used a V-cycle multigrid method to get the fourth-order accurate solutions on both the fine and the coarse grids first, and chose the iterative operator with Richardson extrapolation technique to compute the sixth-order accurate solution on the fine grid. (2) Instead of performing a fixed number of multigrid iterations at each level of grid as used in the usual FMG method, a tolerance j related to the relative residual is used on jth grid level (see line 4 in Algorithm 3), which enables us to conveniently calculate the approximate solution with the desired accuracy and keep suitable computational cost.
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