A novel class of collocation methods based on the weighted integral form of ODEs

Urevc, Janez (author), Starman, Bojan (author), Maček, Andraž (author), Halilovič, Miroslav (author)

In this work, a novel class of collocation methods for numerical integration of ODEs is presented. Methods are derived from the weighted integral form of ODEs by assuming that a polynomial function at individual time increment approximates the solution of the ODE. A distinct feature of the approach, which we demonstrated in this work, is that it allows the increase of accuracy of a method while retaining the number of method coefficients. This is achieved by applying different quadrature rule to the approximation function and the ODE, resulting in different behaviour of a method. Quadrature rules that we examined in this work are the Gauss–Legendre and Lobatto quadrature where several other quadrature rules could further be explored. The approach has also the potential for enhancing the accuracy of the established Runge–Kutta-type methods. We formulated the methods in the form of Butcher tables for convenient implementation. The performance of the new methods is investigated on some well-known stiff, oscillatory and non-linear ODEs from the literature.

• Language: English
• Keywords: collocation methods, numerical integration, ordinary differential equations, stiff systems
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FS - Faculty of Mechanical Engineering
• Publication status: Published
• Publication version: Author Accepted Manuscript
• Year: 2021
• Number of pages: 28 str.
• Numbering: Vol. 40, iss. 4, art. 135
• PID: 20.500.12556/RUL-160915-a914a38b-e543-8a0f-388a-7cbdf5a4d658
• UDC: 517.91(045)
• ISSN on article: 1807-0302
• DOI: 10.1007/s40314-021-01506-6
• COBISS.SI-ID: 62500867
• Publication date in RUL: 05.09.2024

Record is a part of a journal

• Title: Computational & applied mathematics
• Shortened title: Comput. appl. math.
• Publisher: Springer Nature, Sociedade Brasileira de Matemática Aplicada e Computacional
• ISSN: 1807-0302
• COBISS.SI-ID: 520491801

Secondary language

• Language: Slovenian
• Keywords: kolokacijske metode, numerična integracija, navadne diferencialne enačbe, trdi sistemi

Projects

• Funder: ARRS - Slovenian Research Agency
• Project number: P2-0263
• Name: Mehanika v tehniki