In a previous paper, a calculation system for a high-accuracy, high-speed calculation of a one-dimensional (1D) Poisson equation based on the interpolation finite difference method was shown. Spatial high-order finite difference (FD) schemes, including a usual second-order accurate centered space FD scheme, are instantaneously derived on the equally spaced/unequally spaced grid points based on the definition of the Lagrange polynomial function. The upper limit of the higher order FD scheme is not theoretically limited but is studied up to the tenth order, following the previous paper. In the numerical analyses of the 1D Poisson equation published in the previous paper, the FD scheme setting method, SAPI (m), m = 2, 4, …, 10, was defined. Due to specifying the value of m, the setting of FD schemes is uniquely defined. This concept is extended to the numerical analysis of two-dimensional Poisson equations. In this paper, we focus on Poiseuille flows passing through arbitrary cross sections as numerical calculation examples. Over regular and irregular domains, three types of FD methods—(i) forward time explicit method, (ii) time marching successive displacement method, and (iii) alternative direction implicit method—are formulated, and their characteristics of convergence and numerical calculation errors are investigated. The numerical calculation system of the 2D Poisson equation formulated in this paper enables high-accuracy and high-speed calculation by the high-order difference in an arbitrary domain. Especially in the alternative direction implicit method using the band diagonal matrix algorithm, convergence is remarkably accelerated, and high-speed calculation becomes possible.

中文翻译：

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插值有限差分法对二维泊松方程的高阶差分数值分析及计算误差评估

在以前的论文中，显示了一种基于插值有限差分法的高精度，高速一维（1D）泊松方程计算系统。基于拉格朗日多项式函数的定义，在等距/非等距网格点上瞬时导出空间高阶有限差分（FD）方案，包括常用的二阶精确中心空间FD方案。高阶FD方案的上限在理论上不受限制，但在前一论文之后一直进行到十阶研究。在先前发表的一维泊松方程的数值分析中，定义了FD方案设置方法SAPI（m），m = 2、4，…，10。由于指定了m的值，FD方案的设置是唯一定义的。这个概念扩展到二维泊松方程的数值分析。在本文中，我们将集中在通过任意横截面的泊瓦流作为数值计算示例。在规则和不规则域上，制定了三种类型的FD方法：（i）前向时间显式方法，（ii）时间行进连续位移方法和（iii）替代方向隐式方法，以及它们的收敛性和数值计算误差的特征被调查。本文提出的二维泊松方程数值计算系统可以通过任意域中的高阶差分实现高精度和高速计算。特别是在使用带对角矩阵算法的替代方向隐式方法中，