Piecewise Fixed-Step Algorithm for Solution of Ordinary Algebraic-Differential Equations

This paper is dedicated to numerical analysis of ordinary differential equations by means of backward differentiation formulae and numerical differentiation formulae. These algorithms are suitable for stiff tasks. Their considerable improvement, especially efficiency increase has been achieved by means of a piecewise fixed-step method. This approach consists in keeping a step constant within a segment dependent of an order of a differentiation formula. At the end of the segment a local truncation error estimation and a new step prediction is performed from constant step formulae instead of variable step formulae. The concept of the piecewise fixed-step method has been implemented in Matlab platform and tested. Its stability, accuracy, efficiency and numerical complexity has been analyzed theoretically and empirically. Stiff and non-stiff benchmarks have been used for tests.