Finite element θ-schemes for the acoustic wave equation

In this paper, we investigate the stability and convergence of a family of implicit finite difference schemes in time and Galerkin finite element methods in space for the numerical solution of the acoustic wave equation. The schemes cover the classical explicit second-order leapfrog scheme and the fourth-order accurate scheme in time obtained by the modified equation method. We derive general stability conditions for the family of implicit schemes covering some well-known CFL conditions. Optimal error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the L²-norm error over a finite time interval converges optimally as O(h^p+1 Δ+t^s), where p denotes the polynomial degree, s=2 or 4, h the mesh size, and Δt the time step.