We extend and analyze the energy-based discontinuous Galerkin method for second order wave equations on staggered and structured meshes. By combining spatial staggering with local time-stepping near boundaries, the method overcomes the typical numerical stiffness associated with high order piecewise polynomial approximations. In one space dimension with periodic boundary conditions and suitably chosen numerical fluxes, we prove bounds on the spatial operators that establish stability for CFL numbers $c \frac {\Delta t}{h} < C$ independent of order when stability-enhanced explicit time-stepping schemes of matching order are used. For problems on bounded domains and in higher dimensions we demonstrate numerically that one can march explicitly with large time steps at high order temporal and spatial accuracy.
翻译:我们扩展并分析在交错和结构化的中间线上第二波方程式的基于能量的不连续的Galerkin方法。 通过将空间悬浮与当地在近边界附近的时间间隔相结合,该方法克服了与高顺序的片段多面近似相联的典型数字僵硬性。 在一个带有定期边界条件和适当选择的数字通量的空间维度方面,我们证明对空间操作员的界限,这些操作员为CFL数字稳定了 $c\frac = Delta t ⁇ h} < C$ 与使用稳定增强明确时间间隔的匹配顺序计划时的顺序无关。 对于在条框域和更高维度上的问题,我们从数字上表明,在高度时间和空间精确度上,可以明确以大步骤前进。