High Performance Numerical Solution of Partial Differential Equations

A Richardson extrapolation-based sixth-order method is developed for 2D and 3D steady-state equations on uniform grids. Richardson extrapolation is applied to explicitly compute a sixth-order solution on the coarse grid from two fourth-order solutions with different related scale grids. Other computational techniques (i.e., iterative operator based interpolation, multiple coarse grid updating strategy, and completed Richardson extrapolation) are discussed to obtain a higher-order solution on the fine grid. No extra cost is needed to build such kind of multilevel grids if multigrid methods are involved as the solver for the resulting linear systems. For this reason, a multiscale multigrid method is used to achieve both high accuracy and high efficiency computational goals in one framework. Richardson extrapolation-based computation is also extended to solve unsteady-state equations. A higher-order alternating direction implicit method with completed Richardson extrapolation is developed for solving unsteady 2D convection-diffusion equations. The completed Richardson extrapolation is used to improve the accuracy of the solution in spatial and temporal domains simultaneously.