Estimating the Local Error of the Explicit Euler Method 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 limit singular points on integral curves. It is well known that traditional explicit methods for solving the Cauchy problem are inefficient for this class of problems. Implicit methods are much more difficult to use and do not always lead to a result of the desired accuracy. Therefore, along with traditional methods of numerical integration of the Cauchy problem, the 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 present the results of a study of the local error for the numerical solution obtained by the explicit Euler method. An estimate of the 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 bound for the local error is obtained and the efficiency of using the best argument is proved. This is reflected in the decrease of the solution local error for the transformed problem in a neighborhood of limit singular points. The theoretical results are consistent with the numerical solution of an ill-conditioned initial value problem of deformable solid mechanics with one limit singular point.