A novel semi-discrete scheme preserving uniformly exponential stability for an Euler–Bernoulli beam

Abstract

In this paper, a novel space semi-discretized numerical scheme which is based on finite volume method is proposed for approximation of uniformly exponential decay of Euler–Bernoulli beam system, which turns out to be an alternative of finite-difference scheme from order reduction point of view. The new scheme is constructed on equidistant grid points without using any numerical viscosity terms. The uniformly exponential decay is proved by the Lyapunov function method and the energy multiplier technique. With construction of a new gradient recovery function, the numerical solution is proved to be convergent to the (weak) solution of the original continuous system. Compared with the existing literature, the proposed approach has potentially achieved the following objectives: a) It removes the introduction of the numerical viscosity term to achieve uniform convergence; b) It can deal with any type of boundary conditions without help of the spectral analysis which is limited only for some special boundary conditions; c) the convergence proof is simplified significantly with the similar techniques in dealing with the continuous counterpart.