Lyapunov numerical stability of a hyperbolic system of linear balance laws with inhomogeneous coefficients

In the paper, we consider a mixed problem for a class of systems of linear hyperbolic balance laws with inhomogeneous coefficients and with dissipative boundary conditions. The presence of a source term creates additional difficulties for constructing and analyzing the stability (according to Lyapunov) of difference schemes for the numerical calculation of exponentially stable solutions. In the work, we propose an upwind difference splitting scheme for the numerical calculation of stable solutions to the mixed problem for systems of linear hyperbolic balance laws with inhomogeneous coefficients. A discrete analog of the Lyapunov function is constructed and an a priori estimate is obtained for it, which means the exponential stability of the numerical solution.

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