Compact difference schemes for parabolic equations on the base of Runge–Kutta methods
https://doi.org/10.29235/1561-8323-2026-70-1-7-13
Abstract
In this work, for the first time, stable and economical compact finite-difference schemes of order of accuracy 2 + 4 and 4 + 4 are constructed for the simplest parabolic equation, based on the idea of the method of lines and Runge–Kutta methods for solving systems of nonlinear ordinary differential equations. In constructing the computational algorithm, only a three-point stencil is used for approximating the equation by the spatial variable, which allows us to use the well-known tridiagonal matrix algorithm for inverting the matrix in O(N) arithmetic operations, where N is the number of grid points in space. When constructing compact schemes of the same order based on the conventional integro-interpolation method leads only to absolutely unstable algorithms. Results of computational experiments are presented, illustrating the efficiency of the proposed algorithm.
About the Authors
P. P. Matus
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Matus Piotr P. – Corresponding Member, D. Sc. (Physics and Mathematics), Professor, Chief Researcher.
11, Surganov Str., 220072, Minsk
V. T. K. Tuyen
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Tuyen Vo Thi Kim – Ph. D. (Physics and Mathematics), Researcher.
11, Surganov Str., 220072, Minsk
B. V. Faleichik
Belarusian State University
Belarus
Belarus
Faleichik Boris V. – Ph. D. (Physics and Mathematics), Associate Professor.
4, Nezavisimosti Ave., 220030, Minsk
References
1. Matus P. P., Gromyko G. Ph., Utebaev B. D., Tuyen V. T. K. Conservative compact and monotone fourth-order difference schemes for one-dimensional and two-dimensional quasilinear equations. Differential Equations, 2025, vol. 61, no. 8, pp. 1117–1134 (in Russian). https://doi.org/10.7868/s3034503025080097
2. Samarskii A. A. Schemes of high-order accuracy for the multi-dimensional heat conduction equation. USSR Computational Mathematics and Mathematical Physics, 1963, vol. 3, no. 5, pp. 1107–1146. https://doi.org/10.1016/0041-5553(63)90104-6
3. Matus P. P., Gromyko G. Ph., Utebaev B. D. Conservative compact and monotone fourth order difference schemes for quasilinear equations. Doklady Natsional’noi akademii nauk Belarusi = Doklady of the National Academy of Sciences of Belarus, 2024, vol. 68, no. 1, pp. 7–14 (in Russian). https://doi.org/10.29235/1561-8323-2024-68-1-7-14
4. Matus P. P., Tuyen V. T. K., Utebaev B. D. Three-layer compact difference scheme for hyperbolic heat conduction equation. Mathematical Models and Computer Simulations, 2025, vol. 17, no. 6, pp. 675–686. https://doi.org/10.1134/s2070048225700413
5. Shirobokov N. V. Diagonally implicit Runge–Kutta schemes. Computational Mathematics and Mathematical Physics, 2002, vol. 42, no. 7, pp. 974–979.
6. Rogov B. V., Mikhailovskaya M. N. On the convergence of compact difference schemes. Mathematical Models and Computer Simulations, 2009, vol. 1, no. 1, pp. 91–104. https://doi.org/10.1134/S2070048209010104
7. Hartman P. Ordinary Differential Equations. Wiley, 1964.
8. Skvortsov L. M. On implicit Runge–Kutta methods obtained as a result of the inversion of explicit methods. Mathematical Models and Computer Simulations, 2017, vol. 9, pp. 498–510. https://doi.org/10.1134/S2070048217040123
9. Ispolov Yu. G., Postoyalkina E. A., Shabrov N. N. Development of numerical methods and organization of numerical integration algorithms for nonstationary heat conduction problem. Differentsial’nye uravneniya i protsessy upravleniya = Differential Equations and Control Processes, 2002, no. 2, pp. 1–25 (in Russian).
10. Hundsdorfer W., Verwer J. Numerical Solution of Time-Dependent Advection–Diffusion–Reaction. Berlin, Heidelberg, Springer, 2003. 472 p. https://doi.org/10.1007/978-3-662-09017-6
11. Hairer E., Wanner G. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer Berlin, Heidelberg, 1996. https://doi.org/10.1007/978-3-642-05221-7
12. Samarskii A. A., Gulin A.V. Stability of Difference Schemes. Moscow, 1973. 416 p. (in Russian).
13. Wang T. Convergence of an eight-order compact difference scheme for the nonlinear Schrödinger equation. Advances in Numerical Analysis, 2012, vol. 2012, art. 913429. https://doi.org/10.1155/2012/913429
Review
Views:
262
JATS XML
Email this article 