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An efficient optimized adaptive step-size hybrid block method for integrating differential systems

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  • Singh, Gurjinder
  • Garg, Arvind
  • Kanwar, V.
  • Ramos, Higinio

Abstract

This paper deals with the development, analysis and implementation of an optimized hybrid block method having different features, for integrating numerically initial value ordinary differential systems. The hybrid nature of the proposed one-step scheme allows us to bypass the first Dahlquist’s barrier on linear multi-step methods. The theory of interpolation and collocation has been used in the development of the method. We assume an appropriate polynomial representation of the theoretical solution of the problem and consider three off-step points in a one-step block. One of these three off-step points is fixed and the other two off-step points are optimized in order to minimize the local truncation errors of the main method and other additional formula. The resulting scheme is of order five having the property of A-stability. An embedded-type approach is used in order to formulate the proposed method in adaptive form, showing a high efficiency. The adaptive method is tested on well-known differential systems viz. the Robertson’s system, a Gear’s system, a system related with Jacobi elliptic functions, the Brusselator system, and the Van der Pol system, and compared with some well-known numerical codes in the scientific literature.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:362:y:2019:i:c:56
    DOI: 10.1016/j.amc.2019.124567
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    References listed on IDEAS

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    1. Ramos, Higinio & Singh, Gurjinder, 2017. "A tenth order A-stable two-step hybrid block method for solving initial value problems of ODEs," Applied Mathematics and Computation, Elsevier, vol. 310(C), pages 75-88.
    2. Ramos, Higinio & Popescu, Paul, 2018. "How many k-step linear block methods exist and which of them is the most efficient and simplest one?," Applied Mathematics and Computation, Elsevier, vol. 316(C), pages 296-309.
    3. 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.
    4. Ramos, Higinio & Singh, Gurjinder & Kanwar, V. & Bhatia, Saurabh, 2016. "An efficient variable step-size rational Falkner-type method for solving the special second-order IVP," Applied Mathematics and Computation, Elsevier, vol. 291(C), pages 39-51.
    5. T. E. Simos & Jesus Vigo Aguiar, 2001. "A Symmetric High Order Method With Minimal Phase-Lag For The Numerical Solution Of The Schrödinger Equation," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 12(07), pages 1035-1042.
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    Cited by:

    1. 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.
    2. Sriwastav, Nikhil & Barnwal, Amit K. & Ramos, Higinio & Agarwal, Ravi P. & Singh, Mehakpreet, 2024. "Advanced numerical scheme and its convergence analysis for a class of two-point singular boundary value problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 216(C), pages 30-48.

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