A comprehensive high-accuracy numerical calculation system for the 2D Poisson equation by the interpolation finite difference method

Abstract

Abstract To conduct numerical calculation in the finite difference method (FDM), a calculation system should ideally be constructed to have three features: (i) the possibility of correspondence to an arbitrary boundary shape, (ii) high accuracy and (iii) high-speed calculation. In this study, the author has proposed and reported the interpolation FDM (IFDM) as a numerical calculation system with the above three characteristics. In this paper, we especially focus on (ii) high accuracy calculation. Regarding the 1D Poisson equation, the author has already reported on the overall picture of numerical calculations and proposed three schemes for high-accuracy numerical calculations: (i) the SAPI(m) scheme, (ii) SOBI(m) scheme, and (iii) CIFD(m) scheme. (m) denotes the accuracy order, which is usually an even number. Conventionally, high-order accuracy schemes up to the sixth order have been researched and reported, but theoretically, there is no upper accuracy order limit for (m) in these three schemes. However, under double-precision calculations, approximately the 10th-order accuracy (m=10) is the practical upper limit for ensuring high-accuracy calculations, and the calculation resulting in 15 significant digits is defined as the virtual-error zero (VE0) calculation. In the case of the 1D Poisson equation, VE0 calculation is possible in almost any scheme if the forcing term is an analytic function. In this paper, the author extends the above conclusion to the case of the 2D Poisson equation. To go from one-dimensional space to two-dimensional space, we generate a naturally deduced component and an algorithm that must be newly added that solves a problem unique to two-dimensional space, and together these become the IFDM calculation system of the 2D Poisson equation. The numerical calculations confirm that VE0 calculation may be possible even in high-accuracy numerical calculation of the two-dimensional Poisson equation. The above is a conclusion in the regular domain, but it is confirmed that calculations with approximately10 significant digits are also possible in the irregular domain.