АНАЛИЗ ПОГРЕШНОСТИ НЕКОТОРЫХ ЯВНЫХ КОНЕЧНО-РАЗНОСТНЫХ МЕТОДОВ РЕШЕНИЯ ЗАДАЧИ КОШИ НА ПРИМЕРЕ МОДЕЛЬНОГО УРАВНЕНИЯ ДАЛКВИСТА

Abstract

The aim of the paper is to estimate the minimum appropriate number of nodes on a uniform grid (maximum integration step) to obtain a given accuracy for the finite-difference Runge-Kutta methods of the first and second orders of accuracy for the Dahlquist model equation. The error of finite-difference methods is analytically investigated by explicit comparing the values of the exact solutions of the differential and difference Cauchy problems in the nodes of a uniform grid in modulus, and the global error is determined by the maximum of the modules of the local errors on the selected grid. The estimates of the global error are obtained from the inequalities based on the expansions of the functions of the exponent and the logarithm in the Taylor and Mercator series, and clearly depend on the number of nodes of the uniform grid. The bottom of the number of nodes of the uniform grid that is required to have the desirable accuracy to solve the Cauchy problem by above methods is obtained. The obtained estimate of the global error of the direct Euler method for the Dahlquist model equation substantially refines the similar estimate from the paper (Hairer E., and Lubich C. Numerical Solution of Ordinary Differential Equations) and enables us to use an integration step of 1.7 times more in value, keeping the given approximation accuracy. The accuracy order of the finite-difference schemes in the theory of numerical methods for integrating differential equations provides a relationship between the global error of the method and the integration step, however, it does not allow us to directly express the approximation accuracy on the given grid, and therefore, an optimal integration step is most often determined experimentally. The paper studies such a relationship explicitly as a model example and shows one of the possible ways to obtain analytical estimates of the integration step for a given approximation accuracy. A direct study of the global error of finite-difference schemes is important in problems where a trade-off between the approximation accuracy and the complexity (amount of computation) is of importance when the number of grid nodes matters. In this regard, it is of interest to extend similar studies of error estimation to the other finite-difference schemes, namely Runge-Kutta methods of higher orders of accuracy and multistep methods. The results obtained can be useful for solving the tasks of computer modeling and computer-based learning.