Uniform boundary stabilization of the finite difference space discretization of the 1-D wave equation (Q878088)

The authors investigate the classical finite difference approximation of the one-dimen\-sional wave equation with (Dirichlet data at \(x=0\) and) a third-kind boundary condition at \(x=1\) which means dissipation of energy. The classical difference approximation is shown not to reproduce (independently of \(h\)) the exponential energy decay with time. Whereas a compact difference variant of the classical scheme conserves that decay property of the analytical solutions, uniformly in \(h\). In fact, the proposed approximation does apply an \(O(h^2)\) discrete dissipation throughout the interval (0,1). During the proof, several inequalities of independent interest are established.