Nonlinearity difficulty in numerical solving of superstiff Cauchy problems

A lot of schemes has been proposed for solving of stiff Cauchy problems for ordinary differential equations. They are effective on linear and weakly nonlinear systems. The article provides the investigation of behavior of several well-known schemes on strongly nonlinear superstiff problems (including, for example, chemical kinetics problem). Well-known numerical methods are shown to be unreliable for such problems. They require sufficient grid densening in particular critical moments of time though there are no reliable enough procedures to obtain these moments. It has been shown that choosing of time as integration argument leads to difficulty in boundary layer. In case the argument is the integral curve arc length the difficulties occur in transition zone between the boundary layer and the regular solution.