SIAM Journal on Control and Optimization, Vol.58, No.4, 2256-2287, 2020
A NEW SEMIDISCRETIZED ORDER REDUCTION FINITE DIFFERENCE SCHEME FOR UNIFORM APPROXIMATION OF ONE-DIMENSIONAL WAVE EQUATION
In this paper, we propose a novel space semidiscretized finite difference scheme for approximation of the one-dimensional wave equation under boundary feedback. This scheme, referred to as the order reduction finite difference scheme, does not use numerical viscosity and yet preserves the uniform exponential stability. The paper consists of four parts. In the first part, the original wave equation is first transformed by order reduction into an equivalent system. A standard semidiscretized finite difference scheme is then constructed for the equivalent system. It is shown that the semidiscretized scheme is second-order convergent and that the discretized energy converges to the continuous energy. Very unexpectedly, the discretized energy also preserves uniformly exponential decay. In the second part, an order reduction finite difference scheme for the original system is derived directly from the discrete scheme developed in the first part. The uniformly exponential decay, convergence of the solutions, as well as uniform convergence of the discretized energy are established for the original system. In the third part, we develop the uniform observability of the semidiscretized system and the uniform controllability of the Hilbert uniqueness method controls. Finally, in the last part, under a different implicit finite difference scheme for time, two numerical experiments are conducted to show that the proposed implicit difference schemes preserve the uniformly exponential decay.