Algorithmic implementation of an exact three-point difference scheme for a certain class of singular Sturm–Liouville problems

In this article, we present a new algorithmic implementation of exact three-point difference schemes for a certain class of singular Sturm–Liouville problems. We demonstrate that computing the coefficients of the exact scheme at any grid node $x_j$ requires solving two auxiliary Cauchy problems for the second-order linear ordinary differential equations: one problem on the interval $[x_{j-1},x_{j}]$ (forward) and one problem on the interval $[x_{j},x_{j+1}]$ (backward). We have also proven the coefficient stability theorem for the exact three-point difference scheme.

singular Sturm–Liouville problem
exact three-point difference scheme
coefficient stability
  1. Tikhonov A. N., Samarskii A. A.  Homogeneous difference schemes.  USSR Computational Mathematics and Mathematical Physics.  1 (1), 5–67 (1962).
  2. Tikhonov A. N., Samarskii A. A.  Homogeneous difference schemes of a high degree of accuracy on non-uniform nets.  USSR Computational Mathematics and Mathematical Physics.  1 (3), 465–486 (1962).
  3. Prikazchikov V. G.  High-accuracy homogeneous difference schemes for the Sturm–Liouville problem.  USSR Computational Mathematics and Mathematical Physics.  9 (2), 76–106 (1969).
  4. Makarov V. L., Gavrilyuk I. P., Luzhnykh V. M.  Exact and truncated difference schemes for a class of degenerate Sturm–Liouville problems.  Differentsial'nye Uravneniya.  16 (7), 1265–1275 (1980).
  5. Gavrilyuk I. P., Hermann M., Makarov V. L., Kutniv M. V.  Exact and truncated difference schemes for boundary value ODEs.  International Series of Numerical Mathematics.  159.  Birkhäuser Basel (2011).
  6. Kunynets A. V., Kutniv M. V., Khomenko N. V.  Algorithmic realization of an exact three-point difference scheme for the Sturm–Liouville problem.  Journal of Mathematical Sciences.  270 (1), 39–58 (2023).
  7. Kunynets A. V., Kutniv M. V., Khomenko N. V.  Three-point difference schemes of high accuracy order for Sturm–Liouville problem.  Mathematical Methods and Physicomechanical Fields.  63 (4), 54–62 (2020), (in Ukrainian).
  8. Samarskii A. A., Makarov V. L.  On the realization of exact three-point difference schemes for second-order ordinary differential equations with piecewise smooth coefficients.  Soviet Mathematics. Doklady.  41 (3), 463–467 (1990).
  9. Samarskii A. A., Makarov V. L.  Realization of exact three-point difference schemes for second-order ordinary differential equations with piecewise smooth coefficients.  Differ. Equat.  26 (7), 922–930 (1991).
  10. Samarskii A. A.  Introduction to the Theory of Difference Schemes.  Nauka, Moscow (1971).
  11. Hartman Ph.  Ordinary differential equations.  John Wiley & Sons, New York (1964).