Uniform Stabilization and Numerical Analysis of a Thermoviscoelastic System with the Guyer-Krumhansl Model

Abstract

In this work, we investigate a thermoviscoelastic system governed by the Guyer-Krumhansl model. The system is free from the paradox of infinite heat propagation speed and, furthermore, is more suitable for modeling complex problems involving heterogeneous materials on a macroscale and at room temperature. Firstly, we establish the well-posedness of the system using the theory of semigroups of linear operators, and then we prove uniform exponential decay with respect to a given physical parameter using the multiplier method. Subsequently, we discretize the system and propose a monotone and consistent numerical scheme using finite differences. The convergence of the numerical solution is proven by the Lax equivalence theorem. Finally, we present numerical experiments using MATLAB to demonstrate the accuracy and efficiency of the scheme that reproduce the theoretical results.