Interpolation numerical integration method for extremely high-accuracy calculation of ordinary differential equations

Abstract

Abstract There are three types of problems that arise in the numerical analysis of differential equations: (i) the initial value problem (IVP), (ii) the boundary value problem (BVP), and (iii) IVP + BVP. In general, it is natural that (i) and (ii) are thoroughly examined before the study of (iii). Thus far, it has been shown that the BVP of one-dimensional (1D) Poisson equations can be solved with extremely high accuracy and speed using the interpolation finite difference method (IFDM). Furthermore, it has been found that a high-accuracy numerical calculation system for the IVP can be constructed by applying the polynomial interpolation method, as in the case of the BVP of the 1D Poisson equation. This method is called the interpolation numerical integration method (INIM). Numerical integration is possible with an arbitrary accuracy order by specifying the degree of the interpolation polynomial for representing the integrand. The proposed calculation method enables numerical calculations that seem to have no error. IVPs are usually calculated by the fourth-order accuracy Runge‒Kutta method, which is often sufficient for engineering purposes. However, in this paper, we report that a numerical algorithm that agrees with the exact solution up to 15 significant digits has theoretical value.