Finite Element $θ$-Schemes for the Acoustic Wave Equation (Advances in Applied Mathematics and Mechanics, 2018)
In this paper, we investigate the stability and convergence of a family ofimplicit finite difference schemes in time and Galerkin finite element methods inspace for the numerical solution of the acoustic wave equation. The schemes coverthe classical explicit second-order leapfrog scheme and the fourth-order accuratescheme in time obtained by the modified equation method. We derive general stabilityconditions for the family of implicit schemes covering some well-known CFLconditions. 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 intervalconverges optimally as $\mathcal{O}(h^{p+1}+∆t^s)$, where $p$ denotes the polynomial degree, $s$=2 or 4, $h$ the mesh size, and $∆t$ the time step.