Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Janez Urevc and Miroslav Halilovič
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Janez Urevc: Faculty of Mechanical Engineering, University of Ljubljana, 1000 Ljubljana, Slovenia
Miroslav Halilovič: Faculty of Mechanical Engineering, University of Ljubljana, 1000 Ljubljana, Slovenia

Mathematics, 2021, vol. 9, issue 2, 1-21

Abstract: In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

Keywords: collocation methods; Runge–Kutta methods; numerical integration; ordinary differential equations; stiff systems (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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