Exponential stability of an upwind difference scheme for a quasilinear hyperbolic system with nonlinear boundary conditions

Exponential stability of an upwind difference scheme for a quasilinear hyperbolic system with nonlinear boundary conditions

Authors

  • R.Aloev

Abstract

Annotation. In this paper, we numerically study the stability of a difference scheme for a mixed problem
for a quasilinear hyperbolic system of equations. A difference scheme was constructed, and a theorem on
the stability of the numerical solution was proved.

References

Б. П. Демидович, Лекции по математической теории устойчивости, Москва, изд-во «Наука», 1967.

Д. И. Мартынюк, Лекции по качественной теории разностных уравнений, Киев, «Наукова Думка», 1972.

Ю. Г. Евтушенко, В. Г. Жадан, Применение метода функций Ляпунова для исследования сходимости численных методов, ЖВММФ, 15, № 1 (1975), 101—112.

В. И. Зубов, Методы Ляпунова и их применение, Ленинград, изд-во ЛГУ, 1957.

Banda, M. K., & Herty, M. (2013). Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws. Math. Control Relat. Fields,3 (2), 121–142.

Gottlich, S., & Schillen, P. (2017). Numerical discretization of boundary control problems for systems of balance laws: Feedback stabilisation. European Journal of Control , 35 , 11–18.

Weldegiyorgis, G. (2017). Numerical stabilization with boundary controls for hyperbolic systems of balance laws. (Preprint on webpage at http://hdl.handle.net/2263/60870, Accessed:2017 Aug 19).

Mapundi K. Bandaa and Gediyon Y. Weldegiyorgisa. Numerical boundary feedback stabilisation of non-uniform hyperbolic systems of balance laws. International Journal of Control Volume 93, 2020 - Issue 6, Pages 1428-1441.

Jean-Michel Coron, Brigitte d‘Andrea-Novel and Georges Bastin. A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws. IEEE Transactions on Automatic Control · February 2007.

Самарский А.А. Теория разностных схем. Москва. «Наука». 1989 г., стр. 109.

Aloev R., Berdyshev A., Bliyeva D., Dadabayev S., Baishemirov Z .Stability Analysis of an Upwind Difference Splitting Scheme for Two-Dimensional Saint–Venant Equations. Symmetry, 2022-09, journalarticle DOI: 10.3390/sym14101986. Source: Multidisciplinary Digital Publishing Institute.

Aloev R.D., Dadabaev S.U. Stability of the upwind difference splitting scheme for symmetric thyperbolic systems with constant coefficients. Results in Applied Mathematics. 2022 | journal-article. DOI: 10.1016/j.rinam.2022.100298. EID: 2-s2.0-85131461551. Part of ISSN: 25900374.

Aloev R.D., Hudayberganov M.U. A Discrete Analogue of the Lyapunov Function for Hyperbolic Systems. Journal of Mathematical Sciences (United States).2022, journal-article.DOI: 10.1007/s10958-022-06028-y. EID: 2-s2.0-85135683283. Part of ISSN: 15738795 10723374.

Aloev R.D., Eshkuvatov Z.K., Khudoyberganov M.U., Nematova D.E. The Difference Splitting Scheme for n-Dimensional Hyperbolic Systems. Malaysian Journal of Mathematical Sciences. 2022 | journal-article. EID: 2-s2.0-85130020938. Part of ISSN: 18238343.

Aloev R., Berdyshev A., Akbarova, A., Baishemirov, Z. Development of an algorithm for calculating stable solutions of the saint-venant equation using an upwind implicit difference scheme. EasternEuropean Journal of Enterprise Technologies 2021 | journal-article. DOI: 10.15587/1729-4061.2021.239148. EID: 2-s2.0-85116525899. Part of ISSN: 17294061 17293774.

Published

2024-06-07

How to Cite

Aloev, R. (2024). Exponential stability of an upwind difference scheme for a quasilinear hyperbolic system with nonlinear boundary conditions: Exponential stability of an upwind difference scheme for a quasilinear hyperbolic system with nonlinear boundary conditions. MODERN PROBLEMS AND PROSPECTS OF APPLIED MATHEMATICS, 1(01). Retrieved from https://ojs.qarshidu.uz/index.php/mp/article/view/496

Issue

Section

Computational and discrete mathematics