Global'no ustoychivye raznostnye skhemy dlya uravneniya fishera
- Authors: Matus P.P1,2, Pylak D.2
-
Affiliations:
- Institute of Mathematics, National Academy of Sciences of Belarus
- The John Paul Catholic University of Lublin
- Issue: Vol 59, No 7 (2023)
- Pages: 960-967
- Section: Articles
- URL: https://journals.rcsi.science/0374-0641/article/view/141741
- DOI: https://doi.org/10.31857/S0374064123070099
- EDN: https://elibrary.ru/GUZVVS
- ID: 141741
Cite item
Abstract
Unconditionally monotone and globally stable difference schemes for the Fisher equation are constructed and investigated. It is shown that for a certain choice of input data, these schemes inherit the main property of a stable solution of the differential problem. The unconditional monotonicity of the difference schemes under consideration is proved, and an a priori estimate for the difference solution in the uniform norm is obtained. The stable behavior of the difference solution in the nonlinear case is proved under strict constraints on the input data. The results obtained are generalized to multidimensional equations, for the approximation of which economical difference schemes are used.
About the authors
P. P Matus
Institute of Mathematics, National Academy of Sciences of Belarus; The John Paul Catholic University of Lublin
Email: piotr.p.matus@gmail.com
Minsk, 220072, Belarus; Lublin, 20-950, Poland
D. Pylak
The John Paul Catholic University of Lublin
Author for correspondence.
Email: dorotab@kul.pl
Lublin, 20-950, Poland
References
- Колмогоров А.Н., Петровский И.Г., Пискунов Н.С. Исследование уравнения диффузии, соединённой с возрастанием количества вещества и его применение к одной биологической проблеме // Бюлл. Московского гос. ун-та. Секция А. 1937. Т. 1. № 6. С. 1-25.
- Fisher R.A. The wave of advance of advantageous genes // Ann. Hum. Genetic. 1937. V. 7. № 4. P. 353-369.
- Murray J.D. Mathematical Biology: an Introduction. Berlin, 2001.
- Самарский А.А. Теория разностных схем. М., 1989.
- Matus P., Hieu L.M., Vulkov L.G. Analysis of second order difference schemes on nonuniform grids for quasilinear parabolic equations // J. of Comput. and Appl. Math. 2017. V. 310. P. 186-199.
- Годунов С.К. Разностный метод численного расчёта разрывных решений уравнений гидродинамики // Мат. сб. 1959. Т. 47 (89). № 3. С. 271-306.
- Matus P. The maximum principle and some of its applications// Comput. Methods Appl. Math. 2002. V. 2. P. 50-91.
- Matus P., Lemeshevsky S. Stability and monotonicity of difference schemes for nonlinear scalar conservation laws and multidimensional quasi-linear parabolic equations // Comput. Methods Appl. Math. 2009. V. 9. № 3. P. 253-280.
- Матус П.П., Утебаев Б.Д. Компактные и монотонные разностные схемы для параболических уравнений // Мат. моделирование. 2021. Т. 33. № 4. С. 60-78.
- Godlewski E., Raviart P.-A. Hyperbolic Systems of Conservation Law. Paris, 1991.
- Матус П.П., Ирхин В.А., Лапиньска-Хщонович М., Лемешевский С.В. О точных разностных схемах для гиперболических и параболических уравнений // Дифференц. уравнения. 2007. Т. 43. № 7. С. 978-986.
- Lemeshevsky S., Matus P., Poliakov D. Exact Finite-Difference Schemes. Berlin, 2016.
- Jovanovic B., Lemeshevsky S., Matus P. On the stability of differential-operator equations and operator-difference schemes as $t oinfty$ // Comput. Methods Appl. Math. 2002. V. 2. № 2. P. 153-170.