Nonlinearity problem in the numerical solution of superstiff Cauchy problems

Abstract

For the numerical solution of Cauchy stiff initial problems, many schemes have been proposed for ordinary differential equation systems. They work well on linear and weakly nonlinear problems. The article presents a study of a number of well-known schemes on nonlinear problems (which include, for example, the problem of chemical kinetics). It is shown that on these problems, the known numerical methods are unreliable. They require a sufficient step reducing at some critical moments, and to determine these moments, sufficiently reliable algorithms have not been developed. It is shown that in the choice of time as an argument, the difficulty is associated with the boundary layer. If the length of the integral curve arc is taken as an argument, difficulties are caused by the transition zone between the boundary layer and regular solution.