A high-order space–time ultra-weak discontinuous Galerkin method for the second-order wave equation in one space dimension

In this paper, we present and analyze a new space–time ultra-weak discontinuous Galerkin (UWDG) finite element method for the second-order wave equation in one space dimension. The UWDG finite element approximations are used in space variable and also for the temporal approximation. The space–time UWDG discretization is presented in detail, including the definition of the numerical fluxes, which are necessary to obtain optimal error estimates. The proposed scheme can be made arbitrarily high-order accurate in both space and time. The error estimates of the presented semi-discrete and fully-discrete schemes are both analyzed. Several numerical examples are provided to confirm the theoretical results.