Efficient k -Step Linear Block Methods to Solve Second Order Initial Value Problems Directly

Higinio Ramos, Samuel N. Jator and Mark I. Modebei
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Higinio Ramos: Department of Applied Mathematics, University of Salamanca, 37008 Salamanca, Spain
Samuel N. Jator: Department of Mathematics and Statistics, Austin Peay State University Clarksville, Clarksville, TN 37044, USA
Mark I. Modebei: Department of Mathematics Programme, National Mathematical Centre, Abuja 900211, Nigeria

Mathematics, 2020, vol. 8, issue 10, 1-17

Abstract: There are dozens of block methods in literature intended for solving second order initial-value problems. This article aimed at the analysis of the efficiency of k -step block methods for directly solving general second-order initial-value problems. Each of these methods consists of a set of 2 k multi-step formulas (although we will see that this number can be reduced to k + 1 in case of a special equation) that provides approximate solutions at k grid points at once. The usual way to obtain these formulas is by using collocation and interpolation at different points, which are not all necessarily in the mesh (it may also be considered intra-step or off-step points). An important issue is that for each k , all of them are essentially the same method, although they can adopt different formulations. Nevertheless, the performance of those formulations is not the same. The analysis of the methods presented give some clues as how to select the most appropriate ones in terms of computational efficiency. The numerical experiments show that using the proposed formulations, the computing time can be reduced to less than half.

Keywords: ordinary differential equations; second-order initial value problems; multistep block methods; computational efficiency (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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