In the paper, we propose a systematic approach to design and investigate the adequacy of the computational models for a mixed dissipative boundary value problem posed for the symmetric t-hyperbolic systems. We consider a two-dimensional linear hyperbolic system with variable coefficients and with the lower order term in dissipative boundary conditions. We construct the difference splitting scheme for the numerical calculation of stable solutions for this system. A discrete analogue of the Lyapunov�s function is constructed for the numerical verification o f stability o f solutions for the considered problem. A priori estimate is obtained for the discrete analogue of the Lyapunov�s function. This estimate allows us to assert the exponential stability of the numerical solution. A theorem on the exponential stability of the solution of the boundary value problem for linear hyperbolic system and on stability of difference splitting scheme in the Sobolev spaces was proved. These stability theorems give us the opportunity to prove the convergence of the numerical solution. � 2019 by authors, all rights reserved.

The present work is devoted to the proof of uniqueness of the solution of the finite elements scheme in the case of variable coefficients. Finite elements method is applied for the numerical solution of the mixed problem for symmetric hyperbolic systems with variable coefficients. Moreover, dissipative boundary conditions and its stability are proved. Finally, numerical example is provided for the two dimensional mixed problem in simply connected region on the regular lattice. Coding is done by DELPHI7.