Exponential Stability of an Upwind Difference Splitting Scheme for Symmetric t-Hyperbolic Systems

Abstract

This article considers mixed problem for 2-dimensional symmetric t-hyperbolic system. This system consists of constant coefficients. We investigated questions of setting mixed problems (problems with initial-boundary conditions) for symmetric t-hyperbolic system. It is known that for a symmetric t-hyperbolic system, the concept of characteristic velocity plays a very important role. The main point of this article is to construct a difference control scheme for these characteristic velocities. Control parameters are introduced to control the characteristic velocities of a symmetric t-hyperbolic system. The value of the parameter, the characteristic velocities can have different signs. We will consider all possible cases. In these cases, it is obvious that a symmetric t-hyperbolic system with n-equations can be in (n+1)(n+1) different states. These states controlled by control parameters. Upwind difference schemes constructed for each of these states depending on the value of characteristic velocities. The suggested difference schemes for the mixed problem were substantiated is exponential stability by the Lyapunov. It will be shown that the difference scheme will be stable if the CFL condition is satisfied and otherwise unstable.