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

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Mathematical Models and Computer Simulations Aims and scope

Abstract

A fundamentally new approach to the numerical solution of the Cauchy problem for ODEs based on polynomials in the form of basic elements is proposed. In contrast to the explicit methods of Runge–Kutta (RK), Adams, and others, the proposed approach can solve stiff problems. The approach is based on an explicit predictor-corrector (PC) 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 a 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 and 10−15). The fifth order of the method is confirmed by the test for a stiff problem and by the results of an analysis of an asymptotically precise error estimate according to the Richardson scheme on a sequence of refining grids.

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ACKNOWLEDGMENTS

The author thanks Jan Busa for carefully reading the paper, discussing it, and the useful remarks.

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  1. Joint Institute for Nuclear Research, Laboratory of Information Technologies, Dubna, Moscow oblast, Russia

    N. D. Dikusar

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  1. N. D. Dikusar
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Correspondence to N. D. Dikusar.

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Dikusar, N.D. Numerical Solution of the Cauchy Problem Based on the Basic Element Method. Math Models Comput Simul 15, 1024–1036 (2023). https://doi.org/10.1134/S2070048223060091

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