The Parametrization of the Cauchy Problem for Nonlinear Differential Equations with Contrast Structures

Abstract

Introduction. The paper provides an analysis of numerical methods for solving the Cauchy problem for nonlinear ordinary differential equations with contrast structures (interior layers). Similar equations simulate various applied problems of hydro- and aeromechanics, chemical kinetics, the theory of catalytic reactions, etc. An analytical solution to these problems is rarely obtained, and numerical procedure is related with significant difficulties associated with ill-conditionality in the neighborhoods of the boundary and interior layers. The aim of the paper is the scope analysis of traditional numerical methods for solving this class problems and approbation of alternative solution methods. Materials and methods. The traditional explicit Euler and fourth-order Runge-Kutta methods, as well as the implicit Euler method with constant and variable step sizes are used for the numerical solution of the Cauchy problem. The method of solution continuation with respect to the best argument is suggested as an alternative to use. The solution continuation method consists in replacing the original argument of the problem with a new one, measured along the integral curve of the problem. The transformation to the best argument allows obtaining the best conditioned Cauchy problem. Results. The computational difficulties arising when solving the equations with contrast structures by traditional explicit and implicit methods are shown on the example of the test problem solution. These difficulties are expressed in a significant decrease of the step size in the neighborhood of the boundary and interior layers. It leads to the increase of the computational time, as well as to the complication of the solving process for super stiff problems. The authenticity of the obtained results is confirmed by the comparison with the analytical solution and the works of other authors. Conclusions. The results of the computational experiment demonstrate the applicability of the traditional methods for solving the Cauchy problem for equations with contrast structures only at low stiffness. In other cases these methods are ineffective. It is shown that the method of solution continuation with respect to the best argument allows eliminating most of the disadvantages inherent to the original problem. It is reflected in decreasing the computational time and in increasing the solution accuracy. Keywords: contrast structures, method of solution continuation, the best argument, illconditionality, the Cauchy problem, ordinary differential equation For citation: Kuznetsov E. B., Leonov S. S., Tsapko E. D. The Parametrization of the Cauchy Problem for Nonlinear Differential Equations with Contrast Structures. Vestnik Mordovskogo universiteta = Mordovia University Bulletin. 2018; 28(4):486–510. DOI: https://doi.org/10.15507/0236-2910.028.201804.486-510 Acknowledgements: This work was supported by the Russian Science Foundation, project no. 18-19-00474.