Kolobov, V. G. A method for increasing the stability of the numerical solution of differential equation systems. (English. Russian original) Zbl 0715.65061 Sov. J. Autom. Inf. Sci. 22, No. 3, 74-81 (1989); translation from Avtomatika 1989, No. 3, 65-71 (1989). The choice of the local mesh size at time \(t-t_ n\) for the difference approximation of the Cauchy problem \(dx/dt=f(x)\), \(x(0)=x_ 0\), \(t\in [0,T]\), \(x\in R^ m\) is considered in order to minimize some functional of the error vector. This functional includes not the square of the error vector but its projection on the hyperplane which is orthogonal to the phase trajectory of the system at \(t=t_ m\). The truncation error of the Euler scheme and machine arithmetic errors are included in the perturbation of f(x). An algorithm is constructed for finding the optimal integration step. The results of computational experiments are presented. Reviewer: A.I.Tolstykh MSC: 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations 65L05 Numerical methods for initial value problems involving ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems Keywords:error optimization; step size control; choice of the local mesh size; Cauchy problem; Euler scheme; machine arithmetic errors; optimal integration step; computational experiments PDFBibTeX XMLCite \textit{V. G. Kolobov}, Sov. J. Autom. Inf. Sci. 22, No. 3, 74--81 (1989; Zbl 0715.65061); translation from Avtomatika 1989, No. 3, 65--71 (1989)