A semi-discrete finite difference method to uniform stabilization of wave equation with local viscosity

In this paper, we propose a completely new and pure semi-finite difference scheme for uniform exponential convergence of approximation of 1-D wave equation with local viscosity damping by a semi-discrete finite difference scheme. It is known that for this partial differential equation system, the continuous system is exponentially stable yet the classical semi-discrete finite difference scheme is not uniformly exponentially stable. This paper adopts an order reduction finite difference scheme to semi-discretize the continuous system. There are at least three apparent advantages for this approach: a) it keeps the simplicity of the finite difference method; b) the uniformly exponentially convergence is preserved; c) the proof of the uniform exponentially stability is analogous to the continuous part. Finally, it is shown that the solution of the discrete system is convergent to the solution of the continuous counterpart and the energy of the discrete system is uniformly convergent to the continuous energy. The method is systematic to be applicable to any order of PDEs. (C) 2020 Elsevier Ltd. All rights reserved.