Numerical solution of the Cauchy problem based on the basic element method

A fundamentally new approach to the numerical solution of the Cauchy problem for ODE based on polynomials in the form of basic elements. In contrast to the explicit methods of Runge-Kutta, Adams and others, proposed approach can solve stiff problems. The approach is based on an explicit “predictor-corrector” scheme. The calculation of the prediction at the next step is carried out using two polynomials of the fifth degree, connected by additional conditions with double reference to the right side of the equation. The error of the method is regulated by the step length $h$ and the control parameter $K$, $0<K<1$. Such a scheme is stable for calculations with extremely small steps ($h=10^{-17}$, $10^{-15}$). The fifth order of the method is confirmed by the test for the stiff problem, also by the results of an analysis of an asymptotically precise error estimate according to the Richardson scheme on a sequence of shredding grids.