Optimal Order Error Estimates for Discontinuous Galerkin Methods for the Wave Equation

In this paper, we derive optimal order error estimates for spatially semi-discrete and fully discrete schemes to numerically solve the second-order wave equation. The numerical schemes are constructed with the discontinuous Galerkin (DG) discretization for the spatial variable and the centered second-order finite difference approximation for the temporal variable. Under appropriate regularity assumptions on the solution, the schemes are shown to enjoy the optimal order error bounds in terms of both the spatial mesh-size and the time-step. In Grote and Schötzau (J Sci Comput 40:257–272, 2009), a fully discrete DG scheme is studied with an explicit finite difference temporal discretization where a CFL condition is required on the mesh-size and the time-step, and optimal order error estimates are derived in the L2(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(\Omega )$$\end{document}-norm. In comparison, for our fully discrete DG schemes, we do not require a CFL condition on the mesh-size and the time-step, and our optimal order error estimates are derived for the H1(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1(\Omega )$$\end{document}-like norm and the L2(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(\Omega )$$\end{document} norm. Numerical simulation results are reported to illustrate theoretically predicted convergence orders in the H1(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1(\Omega )$$\end{document} and L2(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(\Omega )$$\end{document} norms.