Efficient Numerical Integration Methods for the Cauchy Problem for Stiff Systems of Ordinary Differential Equations

Abstract

The notion of stiffness of a system of ordinary differential equations is refined. The main difficulties encountered when solving the Cauchy problem for stiff systems are indicated. The advantages of switching to a new argument, the integral curve arc length, are demonstrated. Various mesh step selection criteria are discussed, and the integral curve curvature criterion is recommended. The most reliable implicit and explicit schemes suitable for solving stiff problems are presented. A strategy permitting an asymptotically accurate computation of the error of a numerical solution simultaneously with the solution itself is described. An analysis of chemical kinetics of hydrogen combustion in oxygen with allowance for 9 components and 50 reactions between them is provided as an illustration.