Finite Elementθ-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 L^(2)-norm error over a finite time interval converges optimally as O(h^(p+1)+∆t^(s)),where p denotes the polynomial degree,s=2 or 4,h the mesh size,and∆t the time step.