Abstract
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.