Accuracy control in stiff system integration

Previously, for numerical solution of stiff systems of ordinary differential equations, it was proposed to a) use length of the integral curve as the argument and b) choose optimal integration step proportional to $\kappa^{-2/5}$, where $\kappa$ is the curvature of the integral curve. In this work, we construct a test problem in which the exact solution is expressed via elementary functions for both time and arc length arguments. This permitted quantitative comparison of various differential schemes. We show that even explicit Runge-Kutta methods are applicable in calculations with optimal step. The first order scheme provides low accuracy but very high reliability even at enormous stiffness. As order of accuracy increases, reliability of the schemes decreases.
We propose mixed computation strategy. At the first stage, an optimal mesh adapted to solution is built via the first order scheme. At the second stage, this mesh is thickened according to the rule of quasi-uniform meshes splitting and the calculation is done via the scheme with the fourth order of accuracy. The mixed strategy allowed to achieve both good reliability and high accuracy of calculation.