An Algebraic Condition for the Exponential Stability of an Upwind Difference Scheme for Hyperbolic Systems

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Abstract

In the paper, we investigate the question of obtaining the algebraic condition for the exponential stability of the numerical solution of the upwind difference scheme for the mixed problem posed for onedimensional symmetric t-hyperbolic systems with constant coefficients and with dissipative boundary conditions. An a priori estimate for the numerical solution of the boundary-value difference problem is obtained. This estimate allows us to state the exponential stability of the numerical solution. A theorem on the exponential stability of the numerical solution of the boundary-value difference problem is proved. Easily verifiable algebraic conditions for the exponential stability of the numerical solution are given. The convergence of the numerical solution is proved.

About the authors

R. D. Aloev

National University of Uzbekistan

Author for correspondence.
Email: aloevr@mail.ru
Tashkent, Uzbekistan

D. E. Nematova

National University of Uzbekistan

Email: nematova_dilfuza@mail.ru
Tashkent, Uzbekistan

References

  1. Алаев Р. Д., Худайберганов М. У. Дискретный аналог функции Ляпунова для гиперболических систем// Соврем. мат. Фундам. направл. - 2018. -64, № 4. - С. 591-602.
  2. Блохин А. М., Алаев Р. Д. Интегралы энергии и их приложения к исследованию устойчивости разностных схем. - Новосибирск: Изд-во Новосибирского гос. ун-та, 1993.
  3. Го дунов С. К. Уравнения математической физики. - М.: Наука, 1979.
  4. Aloev R. D., Blokhin A. M., Hudayberganov M. U. One class of stable difference schemes for hyperbolic system// Am. J. Numer. Anal. - 2014. -2, № 1. - С. 85-89.
  5. Aloev R. D., Davlatov Sh. O., Eshkuvatov Z. K., Nik Long N. M. A. Uniqueness solution of the finite elements scheme for symmetric hyperbolic systems with variable coefficients// Malays. J. Math. Sci. - 2016. -10. - С. 49-60.
  6. Aloev R. D., Eshkuvatov Z. K., Davlatov Sh. O., Nik Long N. M. A. Sufficient condition of stability of finite element method for symmetric t-hyperbolic systems with constant coefficients// Comput. Math. Appl. - 2014. -68. - С. 1194-1204.
  7. Aloev R. D., Eshkuvatov Z. K., Khudoyberganov M. U., Nematova D. E. The difference splitting scheme for hyperbolic systems with variable coefficients// Math. Statist. - 2019. - 7. - С. 82-89.
  8. Aloev R. D., Eshkuvatov Z. K., Khudayberganov M. U., Nik Long N. M. A. A discrete analogue of energy integral for a difference scheme for quasilinear hyperbolic systems// Appl. Math. - 2018. -9. - С. 789- 805.
  9. Aloev R. D., Khudoyberganov M. U., Blokhin A. M. Construction and research of adequate computational models for quasilinear hyperbolic systems// Numer. Algebra Control Optim. - 2018. -8, № 3. - С. 287- 299.
  10. Bastin G., Coron J.-M. Stability and boundary stabilization of 1-D hyperbolic systems. - Basel: Birkhauser,¨ 2016.
  11. Gottlich¨ S., Schillen P. Numerical Discretization of Boundary Control Problems for Systems of Balance Laws: Feedback Stabilization// Eur. J. Control. - 2017. - 35. - С. 11-18.

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