Higher order stabilised explicit Adams-type methods with damping

Abstract

In this paper we continue the study of explicit Adams-type methods with an extended stability interval represented for the first time in the previous article of the authors in «Journal of the Belarusian State University. Mathematics and Informatics» (2021, No. 2). Such methods require only one calculation of f at each step, but at the same time, they have much longer stability intervals than their classical counterparts. The aim of this work is to construct damped modifications of the methods with an extended stability interval of second order and higher and to test their ability to solve stiff systems of ordinary differential equations. In order to extend the stability regions along the real axis, we propose a general optimisation procedure based on grid search with a progressive increase in the damping parameter. We construct several methods of second, third and fourth orders, describe the realisation of the adaptive choice of the integration step, and represent the results of the comparative numerical experiments.