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Three-Point Difference Schemes of High Order of Accuracy for the Sturm–Liouville Problem

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For the Sturm–Liouville problem, we construct three-point difference schemes of high order of accuracy on a nonuniform grid. The proposed difference schemes for each node of the grid x j , j = 1,2,…,N − 1, require solving of two Cauchy problems for the second-order linear ordinary differential equations on the segments [x j−1, x j] (forward) and [x j , x j+1] (backward) carried out for a single step by using an arbitrary one-step method: either the Taylor series expansion or the Runge–Kutta method of the order of accuracy \( \overline{n} \) = 2[(n +1)/2] (n is a positive integer and [ · ] is the integral part of a number). We estimated the accuracy of three-point difference schemes and developed an algorithm for finding their solution. We also present the results of numerical experiments carried out to confirm our theoretical conclusions.

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Authors and Affiliations

  1. Lviv Polytechnic National University, Lviv, Ukraine

    A. V. Kunynets & M. V. Kutniv

  2. Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine

    N. V. Khomenko

  3. Rzeszów University of Technology, Rzeszów, Poland

    M. V. Kutniv

Authors
  1. A. V. Kunynets
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  2. M. V. Kutniv
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  3. N. V. Khomenko
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Correspondence to M. V. Kutniv.

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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 63, No. 4, pp. 54–62, October–December, 2020.

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Kunynets, A.V., Kutniv, M.V. & Khomenko, N.V. Three-Point Difference Schemes of High Order of Accuracy for the Sturm–Liouville Problem. J Math Sci 273, 948–959 (2023). https://doi.org/10.1007/s10958-023-06556-1

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