Karaa, Samir Finite element \(\theta\)-schemes for the acoustic wave equation. (English) Zbl 1262.65131 Adv. Appl. Math. Mech. 3, No. 2, 181-203 (2011). Summary: 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 \(\mathcal{O}(h^{p+1} + \Delta t^s)\), where \(p\) denotes the polynomial degree, \(s=2\) or \(s=\)4, \(h\) the mesh size, and \(\Delta t\) the time step. Cited in 8 Documents MSC: 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs Keywords:finite element methods; discontinuous Galerkin methods; wave equation; implicit methods; energy method; stability condition; optimal error estimates × Cite Format Result Cite Review PDF