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Dmitry N. Chernoivan, South Russian state Polytechnic University (NPI), Novocherkassk, Russia, investorsan100@gmail.com
Polina B. Seredina, South Russian state Polytechnic University (NPI), Novocherkassk, Russia, npi_pm@mail.ru

The article describes the solution of the problem of fundamental instability of the known interval and bilateral methods for the numerical solution of odes at large intervals of integration. The problem is related to the exponential growth of the error and a very rigid restriction on the length of the integration interval. We consider examples illustrating the instability of the known two-sided methods (the classical interval Moore method of the first order, the two-sided B. S. Dobronets method of arbitrarily high order accuracy) at large integration intervals.
The paper deals with two-sided methods of solving the Cauchy problem, effective in the case of large intervals of integration. According to the characteristics of convergence and stability for the case of nonlinear problems, the proposed methods are significantly superior to known analogues with almost equal order of computational cost at the integration step. Relevant theoretical estimates and examples are provided to confirm the effectiveness of the proposed bilateral methods. The developed methods are divided into the following groups. The methods of the first group are based on a posteriori error estimation of a certain real finite-difference method. Qualitative improvement of the convergence and stability of the two-sided method is achieved by using majorant estimates of the norm of grid green functions corresponding to the scheme of the real method used.
When using two-sided methods of the second group, it does not matter in principle, what method is the approximate solution, the error of which is subject to a posteriori evaluation. Majorants are also used to find estimates for the norm of the green function, but not the grid one, but the continuous linearized problem. To calculate majorants for green’s functions of a linear problem, a corresponding two-sided method is used, which complicates the algorithm of calculations. Therefore, the use of methods of the second group is advisable in cases where the approximate solution is obtained by an unknown numerical method or by experimental measurements.

Keywords: two-sided method, interval method, large gaps, instability, errors, guaranteed accuracy.


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Information about authors:
Dmitry N. Chernoivan, South-Russian state Polytechnic University (NPI), master, Novocherkassk, Russia
Polina B. Seredina, South-Russian state Polytechnic University (NPI), bachelor’s degree, Novocherkassk, Russia