An Explicit–Implicit Upwind Difference Splitting Scheme in Directions for a Mixed Boundary Control Problem for a Two-Dimensional Symmetric t-Hyperbolic System
Abstract
:1. Introduction
2. Proposed Methods
2.1. Problem Statement and Preliminary Information
2.2. Explicit–Implicit Upwind Difference Splitting Scheme
3. Results
- for
- for .
- (1)
- Case I:
- (2)
- Case II:
4. Discussion
- To solve the mixed problem for a linear two-dimensional symmetric hyperbolic system with dissipative boundary conditions incorporating control functions, a new explicit–implicit direction splitting method has been introduced.
- Leveraging the Lyapunov stability theory, we have proven a theorem on the exponential stability of the numerical solution of the upwind difference scheme split across spatial variables. Additionally, a discrete Lyapunov function for the numerical solution of the initial-boundary difference problem has been constructed. We derived sufficient algebraic conditions for the exponential stability of the numerical solution.
- We obtained an a priori estimate for the numerical solution of the initial-boundary difference problem, enabling us to gauge the numerical solution using initial data functions and control functions in boundary conditions. This estimation confirms the continuous dependence of the numerical solution on control functions in boundary conditions, thus allowing control over the hyperbolic system.
Author Contributions
Funding
Institutional Review Board Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Berdyshev, A.; Aloev, R.; Abdiramanov, Z.; Ovlayeva, M. An Explicit–Implicit Upwind Difference Splitting Scheme in Directions for a Mixed Boundary Control Problem for a Two-Dimensional Symmetric t-Hyperbolic System. Symmetry 2023, 15, 1863. https://doi.org/10.3390/sym15101863
Berdyshev A, Aloev R, Abdiramanov Z, Ovlayeva M. An Explicit–Implicit Upwind Difference Splitting Scheme in Directions for a Mixed Boundary Control Problem for a Two-Dimensional Symmetric t-Hyperbolic System. Symmetry. 2023; 15(10):1863. https://doi.org/10.3390/sym15101863
Chicago/Turabian StyleBerdyshev, Abdumauvlen, Rakhmatillo Aloev, Zhanars Abdiramanov, and Mohinur Ovlayeva. 2023. "An Explicit–Implicit Upwind Difference Splitting Scheme in Directions for a Mixed Boundary Control Problem for a Two-Dimensional Symmetric t-Hyperbolic System" Symmetry 15, no. 10: 1863. https://doi.org/10.3390/sym15101863
APA StyleBerdyshev, A., Aloev, R., Abdiramanov, Z., & Ovlayeva, M. (2023). An Explicit–Implicit Upwind Difference Splitting Scheme in Directions for a Mixed Boundary Control Problem for a Two-Dimensional Symmetric t-Hyperbolic System. Symmetry, 15(10), 1863. https://doi.org/10.3390/sym15101863