Mesh step selection based on curvature for stiff Cauchy problems

A new method of automatic step construction is proposed for numerical integration of Cauchy problem for ordinary differential equations. The method is based on using of geometrical properties (namely, curvature and slope) of the integration curve. Formulae for curvature of the integration curve are constructed for different choices of the multidimensional space. In two-dimensional case, they are equivalent to well-known formulae but their general multidimensional form is non-trivial.
For the meshes constructed by our method, a procedure of steps splitting is proposed that allows to apply Richardson method and to calculate a posteriori asymptotically precise error estimation for the obtained solutions. There are no such estimations for traditional automatic step selection algorithms. Consequently, the proposed methods sufficiently excel known before algorithms in reliability and trustworthiness. In existing automatic step algorithms, steps can be unexpectedly reduced by 2–4 orders of magnitude without observable reason. This reduces the algorithms' reliability. We have explained the cause of this phenomenon.
The methods proposed in this work are especially effective on highly stiff problems. This is illustrated by numerous calculations.