Polynomial interpolation with repeated Richardson extrapolation to reduce discretization error in CFD

Abstract

The goal of the present study is to present and test a novel numerical procedure for reducing the discretization error associated with several types of variables of interest in basic Computational Fluid Dynamics (CFD) problems. Variables of interest are classified into five types according to their locations on various grids. According to the current literature, Repeated Richardson Extrapolation (RRE) performs well for only one of the five types of variable, i.e., for global variables or those that otherwise have fixed nodal positions on different grids. RRE does not perform well for the remaining four variable types. Because of this limitation, in this work, polynomial interpolation is applied to various numerical solutions obtained on different grids, followed by RRE. Four problems are used to test the proposed procedure, one linear and three non-linear based on the following equations: 1D Poisson, 2D Burgers and 2D Navier–Stokes. These equations are discretized using the Finite Difference method with approximations of second- and fourth-order accuracy and the Finite Volume method with approximations of first- and second-order accuracy. Polynomial interpolation functions for one- and two-dimensional domains are adopted, and optimization techniques are also adopted in some cases. The discretization error is significantly reduced, and the order of accuracy is also increased: for example, based on a second-order scheme with an error of 1.4×10−6, we obtain 2.1×10−27 using six extrapolations on a grid with 1460 elements and an order of accuracy of 14.5. The computational effort (CPU time and memory usage) needed to obtain the solution at a given level of numerical error or using a specific grid is also significantly reduced.