Uniformly absolute exponential and polynomial stability of semi-discrete approximations for wave equation under nonlinear boundary control

In this paper, we apply an order reduction semi-discretization scheme to a wave equation subject to nonlinear boundary control, achieving uniform stability that encompasses both exponential and polynomial stability, with uniformly absolute stability as special cases. Firstly, we showcase that the energy decay rate of the classical finite difference semi-discretized system fails to maintain uniformity with respect to the mesh size, approaching zero as mesh size tends to zero. Next, we propose a finite difference semi-discretization scheme that using the order reduction method and prove that it maintains uniform exponential or polynomial stability with respect to the mesh size. Additionally, we establish the weak convergence of the discrete solution to the continuous solution. Lastly, we conduct numerical experiments to illustrate the non-uniform stabilization of the classical finite difference semi-discretized system in relation to mesh size and validate the effectiveness of the numerical scheme derived from the finite difference method based on order reduction.