Investigation of the stability of the modified difference courant, Isaakson and Rees schemes for quasi-linear hyperbolic systems
Keywords:
mixed problems, quasi-linear hyperbolic equations, model problemAbstract
In the modeling of a number of practical problems, it is brought to the mixed problems imposed on the system of quasi-linear hyperbolic equations. Since there is no exact solution to such problems, they are solved using approximate solution methods. Currently, the theory of differential schemes is widely used in the approximate solution of differential problems. Therefore, the theory of discrete schemes is used to numerically solve the mixed problem presented in this article. To solve any differential problem numerically, it is necessary to prove the correctness of the differential problem under consideration. For this purpose, in the article, an a priori estimate for the solution of the differential problem, with the dissipative condition fulfilled, was obtained for the boundary conditions of the mixed problem placed on the quasi-linear hyperbolic system. After that, in order to construct a discrete analogue of the a priori value obtained for the solution of the differential problem, a differential scheme approximating the quasi-linear hyperbolic system was constructed and the stability of the differential scheme was proved. The stability of the difference scheme approximating the mixed problem set for the differential problem was proved by constructing a discrete analog of the a priori value obtained for the solution of the mixed problem set to the quasi-linear hyperbolic system. A calculation experiment was conducted to numerically solve the model problem with the help of the constructed differential circuit, and the results of the conducted experiment showed the stability of the constructed differential circuit and the obtained solution is close to the exact solution.
References
Blokhin A.M., Alaev R.D. 1993. Energy integrals and their applications to the study of the stability of difference schemes. Novosibirsk: Publishing House of the Novosibirsk University, – 224 p. (in Russian).
Godunov S.K. 1979. Equations of mathematical physics. M.: Nauka, – 372 p.( in Russian)
Khudoyberganov M. Rikhsiboev D., Rashidov J. 2020. About one difference scheme for quasi-linear hyperbolic system // AIP Conf. Proc. – 2021. – Vol. 2365. – DOI:10.1063/5.0057131.
Kulikovskii A.G., Pogorelov N.V., Semenov A Yu. Mathematical Aspects of Numerical Solution of Hyperbolic Systems, Monographs and Surveys in Pure and Applied Mathematics 188, Chapman and Hall/CRC, Boca Raton, FL.
Aloev R.D., Eshkuvatov Z.K., Khudoyberganov M.U., and Nematova D.E. 2019. The difference splitting scheme for hyperbolic systems with variable coefficients, Math. Stat. 7(3), – P. 82–89.
Aloev R.D., Khudoyberganov M.U. 2017. Construction and research of adequate computational models for hyperbolic systems // Abstracts, International conference on control, optimization and differential equations. Malaysia, – 12 p.
Aloev R.D., Khudoyberganov M.U. Investigation of implicit difference schemes for hyperbolic systems. Problems of Computational ana Applied Mathematics, 2017. –4(10). – P. 84–92.
Aloev R.D., Khudoyberganov M.U. 2017. Using an a priori estimate for constructing difference schemes for quasilinear hyperbolic systems. Vestnik NUUz. Tashkent, –2/2. – P. 9-21.
Aloev R.D., Khudoyberganov M.U., Blokhin A.M. Construction and research of adequate computational models for quasi-linear hyperbolic systems // AIMS. Numerical Algebra, Control and Optimization. – 2018. – Vol. 8. – P. 287-299.
Khudoyberganov M.U. 2023. Stability of the difference scheme for the shallow water equation // AIP Conf. Proc. – 2023. – Vol. 2781. – DOI: 10.1063/5.0144765.
