Arc Length and Multilayer Methods for Solving Initial Value Problems for Differential Equations with Contrast Structures

In this paper, we investigate the features of the numerical solution of Cauchy problems for nonlinear differential equations with contrast structures (interior layers). Similar problems arise in the modeling of certain problems of hydrodynamics, chemical kinetics, combustion theory, computational geometry. Analytical solution of problems with contrast structures can be obtained only in particular cases. The numerical solution is also difficult to obtain. This is due to the ill-conditionality of the equations in the neighborhood of the interior and boundary layers. To achieve an acceptable accuracy of the numerical solution, it is necessary to significantly reduce the step size, which leads to an increase of a computational complexity. The disadvantages of using the traditional explicit Euler method and fourth-order Runge-Kutta method, as well as the implicit Euler method with constant and variable step sizes are shown on the example of one test problem with two boundary and one interior layers. Two approaches have been proposed to eliminate the computational disadvantages of traditional methods. As the first method, the best parametrization is applied. This method consists in passing to a new argument measured in the tangent direction along the integral curve of the considered Cauchy problem. The best parametrization allows obtaining the best conditioned Cauchy problem and eliminating the computational difficulties arising in the neighborhood of the interior and boundary layers. The second approach for solving the Cauchy problem is a semi-analytical method developed in the works of Alexander N. Vasilyev and Dmitry A. Tarkhov their apprentice and followers. This method allows obtaining a multilayered functional solution, which can be considered as a type of nonlinear asymptotics. Even at high rigidity, a semi-analytical method allows obtaining acceptable accuracy solution of problems with contrast structures. The analysis of the methods used is carried out. The obtained results are compared with the analytical solution of the considered test problem, as well as with the results of other authors.