The aim of this paper is to apply a high-order discontinuous-in-time scheme to second-order hyperbolic partial differential equations (PDEs). We first discretize the PDEs in time while keeping the spatial differential operators undiscretized. The well-posedness of this semi-discrete scheme is analyzed and a priori error estimates are derived in the energy norm. We then combine this $hp$-version discontinuous Galerkin method for temporal discretization with an $H^1$-conforming finite element approximation for the spatial variables to construct a fully discrete scheme. A prior error estimates are derived both in the energy norm and the $L^2$-norm. Numerical experiments are presented to verify the theoretical results.
翻译:本文的目的是对二阶双曲部分差异方程(PDEs)应用高分级不连续时间办法。我们首先及时将PDEs分解,同时保持空间差分操作器的不分解。分析半分解法的稳妥性,并在能源规范中得出先验误差估计数。然后,我们将这种用于时间离散的美元-转折不连续加勒金方法与用于空间变量以构建完全离散法的空间变量符合的有限要素近似值结合起来。先前的误差估计数在能源规范中和$L2美元-诺尔姆中都得出。提出了数值实验,以核实理论结果。