Solving the Cauchy problem with guaranteed accuracy for stiff systems by the arc length method

The arc length method is an efficient way of solving the Cauchy problem for systems of ordinary differential equations that have areas with large right-hand sides (stiff and ill-conditioned problems). It is shown how to get a posteriori asymptotically exact error estimation for such calculations by using thickening of nets and the Richardson method. The examples of calculations demonstrate that with the transition to the arc length, the greater the stiffness of a problem or its ill conditionality, the larger the gain in the accuracy. This gain can reach many orders of magnitude. It is shown that in order to get reliable results for hyperstiff problems (characterized by large difference in scales of speeds of various processes), it is necessary to make calculations with high digit capacity and/or by an analytical expression of a Jacobian matrix.