Numerical Solution of the Cauchy Problem Based on the Basic Element Method

We offer to your attention a preprint “Numerical Solution of the Cauchy Problem Based on the Basic Element Method” (P5-2021-50) published by the JINR Publishing Department. The author is N. D. Dikusar.

Within the framework of the basic element method (BEM), a fundamentally new approach to the numerical solution of the Cauchy problem for ODE (BEM–PC method) is proposed. In BEM–PC, an explicit scheme of the “predictor–corrector” type is used, conventionally called the “target detection scheme”. Calculation of the prediction for the next step is carried out using two BEM-polynomials of the fifth degree, connected by additional conditions.The first polynomial calculates the “aiming point”, and the second determines the “target coordinate”, i.e. a point close to the exact solution. Such a scheme is stable in calculations with an extremely small step (h = 10−17, 10−15). The method has been verified by a stiff Cauchy problem test and comparisons with errors of popular classical methods. With the help of examples it is shown that the accuracy of the BЕМ–PC method is not worse than the accuracy of the Runge–Kutta method of the fourth order and also of Adams–Bashforth and Fehlberg methods of the fifth order.