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

Author

Listed:
  • Janez Urevc

    (Faculty of Mechanical Engineering, University of Ljubljana, 1000 Ljubljana, Slovenia)

  • Miroslav Halilovič

    (Faculty of Mechanical Engineering, University of Ljubljana, 1000 Ljubljana, Slovenia)

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.

Suggested Citation

  • Janez Urevc & Miroslav Halilovič, 2021. "Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs," Mathematics, MDPI, vol. 9(2), pages 1-21, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:2:p:174-:d:481370
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    References listed on IDEAS

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    1. Singh, Gurjinder & Garg, Arvind & Kanwar, V. & Ramos, Higinio, 2019. "An efficient optimized adaptive step-size hybrid block method for integrating differential systems," Applied Mathematics and Computation, Elsevier, vol. 362(C), pages 1-1.
    2. Ramos, Higinio & Rufai, M.A., 2018. "Third derivative modification of k-step block Falkner methods for the numerical solution of second order initial-value problems," Applied Mathematics and Computation, Elsevier, vol. 333(C), pages 231-245.
    3. Modebei, Mark I. & Adeniyi, Rapheal B. & Jator, Samuel N. & Ramos, Higinio, 2019. "A block hybrid integrator for numerically solving fourth-order Initial Value Problems," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 680-694.
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