Abstract
By using Richardson extrapolation and fourth-order compact finite difference scheme on different scale grids, a sixth-order solution is computed on the coarse grid. Other three techniques are applied to obtain a sixth-order solution on the fine grid, and thus give out three kinds of Richardson extrapolation-based sixth order compact computation methods. By carefully analyzing the truncation errors respectively on 2D Poisson equation, we compare the accuracy of these three sixth order methods theoretically. Numerical results for two test problems are discussed.
Highlights
High order and high efficiency numerical computation for partial differential equations is very important in many scientific and engineering modeling problems
We chose a multiscale multigrid (MSMG) computational framework [22], which uses multigrid methods to speed up the linear system solution, at the same time, involves a multiscale strategy to obtain higher order accurate solution by extrapolating the computed lower order solutions
Compared to the sixth order compact schemes derived by Hermitian polynomial, the Richardson extrapolation-based sixth order compact approximations have many obvious advantages, such as simple stencils, complete compact, easy implementation, suitable for high efficient linear system solvers, etc
Summary
High order and high efficiency numerical computation for partial differential equations is very important in many scientific and engineering modeling problems. Finite difference (and finite element) methods lead to systems with sparse matrices that can be handled by efficient. Taking into account the computational efficiency, we focus on developing numerical algorithms based on high order finite difference methods here. Compared to using straightforward central differences to obtain high order accuracy on a larger stencil which results the increase of bandwidth of the coefficient matrix and rises to a problem at the points close to the boundaries, the HOC schemes only use the center and adjacent points (i.e., a 9-points stencil is used in HOC schemes in 2D) which avoid extra special treatments for those points close to the boundaries and further improve computational efficiency. Various fourth order compact (FOC) finite difference schemes have been developed for Poisson equations, convection-diffusion equations, and Navier-Stokes equations [6] [7] [8] [9]
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