Lyapunov stability of an upwind difference scheme for a mixed problem for a system of two linear hyperbolic equations with variable coefficients
In this paper, the exponential stability of the upwind explicit difference scheme is proved for two linear hyperbolic equations with variable coefficients. The general case of dissipative boundary conditions is considered. An explicit discrete quadratic Lyapunov function is constructed. The aim of this work is to develop effective methods for the numerical solution of a system of two hyperbolic equations with variable coefficients. In this paper, we will use Lyapunov’s direct approach to study the exponential stability of the solution to a system of two hyperbolic equations with variable coefficients. This work is devoted to the study of the issues of exponential stability of the solution of an implicit upwind difference scheme for a system of two hyperbolic equations with variable coefficients. For the numerical value of stable solutions of the mixed problem for a system of two hyperbolic equations with variable coefficients, a discrete analog of the Lyapunov function is constructed and an a priori estimate is obtained for it. The obtained a priori estimate allows one to prove a theorem on the exponential stability of the numerical solution of an implicit upwind difference scheme for a system of two hyperbolic equations with variable coefficients in the corresponding norms. This theorem implies the convergence of a stable numerical solution to a stable solution of a differential problem.