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Highly accurate numerical solutions with repeated Richardson extrapolation for 2D laplace equation

机译：二维Laplace方程的重复Richardson外推法的高精度数值解

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摘要

A theoretical basis is presented for the repeated Richardson extrapolation (RRE) to reduce and estimate the discretization error of numerical solutions for heat conduction. An example application is described for the 2D Laplace equation using the finite difference method, a domain discretized with uniform grids, second-order accurate approximations, several variables of interest, Dirichlet boundary conditions, grids with up to 8,193 × 8,193 nodes, a multigrid method, single, double and quadruple precisions and up to twelve Richardson extrapolations. It was found that: (1) RRE significantly reduces the discretization error (for example, from 2.25E-07 to 3.19E-32 with nine extrapolations and a 1,025 × 1,025 grid, yielding an order of accuracy of 19.1); (2) the Richardson error estimator works for numerical results obtained with RRE; (3) a higher reduction of the discretization error with RRE is achieved by using higher calculation precision, a larger number of extrapolations, a larger number of grids and correct error orders; and (4) to obtain a given value error, much less CPU time and RAM memory are required for the solution with RRE than without it.
机译：为重复的理查森外推法（RRE）提供了一个理论基础，以减少和估计导热数值解的离散化误差。使用有限差分方法，二维均匀离散的域，二阶精确逼近，感兴趣的几个变量，Dirichlet边界条件，具有多达8,193×8,193个节点的网格，多重网格方法描述了二维Laplace方程的示例应用，单精度，双精度和四精度以及最多十二个理查森外推法。发现：（1）RRE显着降低了离散误差（例如，从2.25E-07到3.19E-32，具有9个外推法和1,025×1,025网格，得出的精度为19.1）； （2）Richardson误差估计器适用于通过RRE获得的数值结果； （3）通过使用更高的计算精度，更大数量的外推，更大数量的网格和正确的错误顺序，可以实现使用RRE更高的离散化误差降低； （4）为了获得给定的值误差，使用RRE的解决方案所需的CPU时间和RAM内存要比不使用它的少得多。

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