Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs
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 collocati...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/2/174 |
| _version_ | 1797410856522219520 |
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| author | Janez Urevc Miroslav Halilovič |
| author_facet | Janez Urevc Miroslav Halilovič |
| author_sort | Janez Urevc |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-09T04:36:18Z |
| format | Article |
| id | doaj.art-df0b625598ae47569533d05dce840b53 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T04:36:18Z |
| publishDate | 2021-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-df0b625598ae47569533d05dce840b532023-12-03T13:28:09ZengMDPI AGMathematics2227-73902021-01-019217410.3390/math9020174Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEsJanez Urevc0Miroslav Halilovič1Faculty of Mechanical Engineering, University of Ljubljana, 1000 Ljubljana, SloveniaFaculty of Mechanical Engineering, University of Ljubljana, 1000 Ljubljana, SloveniaIn 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.https://www.mdpi.com/2227-7390/9/2/174collocation methodsRunge–Kutta methodsnumerical integrationordinary differential equationsstiff systems |
| spellingShingle | Janez Urevc Miroslav Halilovič Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs Mathematics collocation methods Runge–Kutta methods numerical integration ordinary differential equations stiff systems |
| title | Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs |
| title_full | Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs |
| title_fullStr | Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs |
| title_full_unstemmed | Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs |
| title_short | Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs |
| title_sort | enhancing accuracy of runge kutta type collocation methods for solving odes |
| topic | collocation methods Runge–Kutta methods numerical integration ordinary differential equations stiff systems |
| url | https://www.mdpi.com/2227-7390/9/2/174 |
| work_keys_str_mv | AT janezurevc enhancingaccuracyofrungekuttatypecollocationmethodsforsolvingodes AT miroslavhalilovic enhancingaccuracyofrungekuttatypecollocationmethodsforsolvingodes |