Exponential decay and numerical treatment for mixture problem with Fourier law and frictional damping

This study investigates a finite difference numerical scheme to analyze the impact of the effects caused by the strong coupling of Fourier’s law on the solutions of the equations of motion of a mixture of two one-dimensional linear isotropic elastic materials with frictional damping. We first prove the existence of solutions and exponential stability. In the sequence, we analyze the semi-discrete problem in finite differences and we use the energy method to prove the exponential stabilization of the corresponding semi-discrete system. The positivity of the numerical energy is also proved, and we present a fully discrete finite difference scheme that combines explicit and implicit integration methods. Finally, numerical simulations are given to confirm the theoretical results and show the efficiency of the proposed scheme.