Numerical Study of Lyapunov Stability of an Upwind Difference Scheme for a Quasilinear Hyperbolic System

This study addresses a mixed problem for a quasilinear system of hyperbolic equations expressed in Riemann invariants, incorporating dissipative nonlinear boundary conditions. A numerical approach is developed through an initial-boundary difference problem utilizing an upwind difference scheme. The stability of nonlinear difference schemes is investigated, with a focus on establishing a sufficient stability criterion based on Lyapunov vector functions. The proposed criterion extends prior theoretical work, where a discrete Lyapunov function was formulated to demonstrate the exponential stability of the steady state for the quasilinear system. Numerical computations for a model problem validate these theoretical findings. The research highlights the potential of adapting the direct Lyapunov method to analyze the stability of nonlinear hyperbolic systems by constructing a positive definite function that exhibits monotonic decay along system solutions.

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