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

Abstract: The paper considers the numerical solution of the Cauchy problem for systems of ordinarydifferential equations. Special attention is paid to problems with limit singular points on integralcurves. It is well known that traditional explicit methods for solving the Cauchy problem areinefficient for this class of problems. Implicit methods are much more difficult to use and do notalways lead to a result of the desired accuracy. Therefore, along with traditional methods ofnumerical integration of the Cauchy problem, the authors use the method of solution continuationwith respect to the best argument (also known as the best parameterization and the arc lengthmethod). The best argument is calculated tangentially along the integral curve of the problemunder consideration. For the Cauchy problems transformed to the best argument, the authorspresent the results of a study of the local error for the numerical solution obtained by the explicitEuler method. An estimate of the local error of the numerical solution for the Cauchy problemtransformed to the best argument is obtained for the explicit Euler method. Using it, an upperbound for the local error is obtained and the efficiency of using the best argument is proved. Thisis reflected in the decrease of the solution local error for the transformed problem ina neighborhood of limit singular points. The theoretical results are consistent with the numericalsolution of an ill-conditioned initial value problem of deformable solid mechanics with one limitsingular point. © Pleiades Publishing, Ltd. 2025.