ON THE ESTIMATION OF THE EXPLICIT EULER METHOD LOCAL ERROR FOR THE NUMERICAL SOLUTION OF THE CAUCHY PROBLEM FOR ORDINARY DIFFERENTIAL EQUATIONS TRANSFORMED TO THE BEST ARGUMENT

The paper considers the numerical solution of the Cauchy problem for systems of ordinary differential equations. Special attention is paid to problems with limiting singular points on integral curves. It is known that traditional explicit methods for solving the Cauchy problem are ineffective for this class of problems. Implicit methods are much more difficult to use and do not always lead to the result of the desired accuracy. Therefore, along with traditional methods of numerical integration of the Cauchy problem authors use the method of solution continuation with respect to the best argument (also known as the best parameterization and the arc length method). The best argument is calculated tangentially along the integral curve of the problem under consideration. For the Cauchy problems transformed to the best argument, the authors in this paper present the results of a study of the local error for the numerical solution obtained by the explicit Euler method. An estimate of the numerical solution local error of the numerical solution for the Cauchy problem transformed to the best argument is obtained for the explicit Euler method. Using it, an upper estimate of the local error was obtained and the effectiveness of using the best argument was proved. This is reflected in a decrease of the solution local error for the transformed problem in the neighborhood of the limiting singular points. The theoretical results are compatible with the numerical solution of the ill-conditioned initial value problem of deformable solid mechanics with one limiting singular point.

explicit Euler method, local error, Cauchy problem, system of ordinary differential equations, best parameterization, limiting singular point