High-accuracy and high-speed calculation of second-order ODE by the interpolation finite difference method

Abstract Numerical calculation of differential equations using the finite difference method (FDM) is entering a new phase. The high-accuracy calculation system of the interpolation FDM (IFDM) enables extremely high-accuracy numerical calculation. One of the higher-order difference schemes of IFDM is defined as the compact interpolation finite difference (CIFD) scheme. According to this finite difference (FD) scheme, it becomes clear that it can be calculated with 15 significant figures in the numerical analysis of the 1D Poisson equation under double precision calculation. Thus, it is possible to perform an apparent error-free calculation. The 1D Poisson equation is a special form of the second-order ordinary differential equation (ODE), but this paper shows that the methods reported thus far are generalized to the general second-order ODE.