An Explicit-Implicit Difference Scheme for a two-Dimensional Linear Hyperbolic System with Dynamic Boundary Conditions
DOI:
https://doi.org/10.71310/pcam.2_72.2026.08Keywords:
hyperbolic system, dynamic boundary condition, difference scheme, directional splitting, explicit–implicit method, exponential stability, CFL conditionAbstract
This paper considers a two-dimensional linear hyperbolic system with dynamic boundary conditions and proposes a difference scheme for its numerical solution. An explicit–implicit directional splitting method is constructed, where discretization is performed explicitly in one direction and implicitly in the other, while preserving the dissipative structure of the boundary conditions. The stability of the scheme is established under the Courant–Friedrichs–Lewy condition and a linear matrix inequality. It is shown that the discrete energy decreases exponentially in time. Numerical experiments confirm the theoretical results, demonstrating monotonic decay of the discrete ????2-norm and good agreement with the exact solution. The proposed method is stable, dissipative, and computationally efficient, and can be effectively applied to two-dimensional hyperbolic systems with dynamic boundary conditions.
References
Coron J.M., Bastin G., d'Andréa Novel B. Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems // SIAM Journal on Control and Optimization. – 2008. – Vol. 47. – Issue 3. – P. 1460-1498. doi: http://dx.doi.org/10.1137/06066363X.
Prieur C., Winkin J., Bastin G. Robust boundary control of systems of conservation laws // Mathematics of Control, Signals, and Systems – 2008. – Vol. 20. – Issue 2. – P. 173-197. doi: http://dx.doi.org/10.1007/s00498-007-0018-0.
Bastin G., Coron J.M., d'Andréa Novel B. On Lyapunov stability of linearized Saint-Venant equations for a sloping channel // Networks and Heterogeneous Media – 2009. – Vol. 4. – Issue 2. – P. 177-187. doi: http://dx.doi.org/10.3934/nhm.2009.4.177.
Castillo F., Witrant E., Dugard L. Contrôle de température dans un flux de Poiseuille // Proceedings of IEEE Conférence Internationale Francophone d'Automatique – 2012. – Grenoble, France.
Castillo F., Witrant E., Prieur C., Dugard L. Dynamic boundary stabilization of hyperbolic systems // Proceedings of the 51st IEEE Conference on Decision and Control (CDC) – 2012. – Maui, HI, USA. doi: http://dx.doi.org/10.1109/CDC.2012.6425820.
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. – Vol. 14. – Article 1986. doi: http://dx.doi.org/10.3390/sym14101986.
Aloev R.D., Dadabaev S.U. Stability of the upwind difference splitting scheme for symmetric-hyperbolic systems with constant coefficients // Results in Applied Mathematics. – 2022. – Vol. 16. – Article 100298. doi: http://dx.doi.org/10.1016/j.rinam.2022.100298.
Aloev R.D., Hudayberganov M.U. A discrete analogue of the Lyapunov function for hyperbolic systems // Journal of Mathematical Sciences – 2022. – Vol. 268. – P. 640-651. doi: http://dx.doi.org/10.1007/s10958-022-06028-y.
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. – Vol. 16. – Issue 3. – P. 421-438.
Aloev R., Khasanov M., Berezovsky A. Construction and research of adequate computational models for quasilinear hyperbolic systems // Numerical Algebra, Control and Optimization. – 2018. – Vol. 8. – Issue 2. – P. 161-177. doi: http://dx.doi.org/10.3934/naco.2018017.
Berdyshev A., Aloev R., Abdiramanov Z., Ovlaeva M. An explicit–implicit upwind difference splitting scheme in directions for a mixed boundary control problem for a two-dimensional symmetric-hyperbolic system // Symmetry. – 2023. – Vol. 15. – Article 1863. doi: http://dx.doi.org/10.3390/sym15101863.
Aloev R.D., Ovlaeva M.Kh. Construction and study of the stability of a difference scheme for a linear hyperbolic system with dynamic boundary // Uzbek Mathematical Journal. – 2023. – Vol. 67. – Issue 2. – P. 17-24. doi: http://dx.doi.org/10.29229/uzmj.2023-2-2.
Aloev R.D., Ovlaeva M.Kh., Nishonaliyeva M. Construction and stability analysis of a difference scheme for a linear hyperbolic system with dynamic boundary conditions // Proceedings of the VII International Scientific and Technical Conference “Problems of Mechanical Engineering”. – 2023. – Omsk, Russia.
Aloev R.D., Ovlaeva M.Kh. Numerical solution of a mixed problem for a hyperbolic system with dynamic boundary conditions // Proceedings of the International Scientific and Practical Conference “System Analysis and Modeling in Economy and International Relations”. – 2026. – Tashkent, Uzbekistan. doi: http://dx.doi.org/10.5281/zenodo.18217074.
Horn R.A., Johnson C.R. Matrix Analysis. – 2nd ed. – 2013. – Cambridge: Cambridge University Press.
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Р.Д. Алоев
This work is licensed under a Creative Commons Attribution 4.0 International License.
