A local discontinuous Galerkin method for the second-order wave equation

In this paper we present new superconvergence results for the local discontinuous Galerkin (LDG) method applied to the second-order scalar wave equation in one space dimension. Numerical experiments show O(hp+1)L2O(hp+1)L2 convergence rate for the LDG solution and O(hp+2) superconvergent solutions at Radau points. More precisely, a local error analysis reveals that, at a fixed time t, the leading terms of the discretization errors for the solution and its derivative using p-degree polynomial approximations are proportional to the (p + 1)-degree right Radau and (p + 1)-degree left Radau polynomials, respectively. Thus, the p-degree LDG solution is O(hp+2) superconvergent at the roots of the (p + 1)-degree right Radau polynomial and the derivative of the LDG solution is O(hp+2) superconvergent at the roots of the (p + 1)-degree left Radau polynomial. These results are used to construct simple, efficient, and asymptotically correct a posteriori error estimates in regions where solutions are smooth. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori errors estimates under mesh refinement.