We develop a hybrid spatial discretization for the wave equation in second order form, based on high-order accurate finite difference methods and discontinuous Galerkin methods. The hybridization combines computational efficiency of finite difference methods on Cartesian grids and geometrical flexibility of discontinuous Galerkin methods on unstructured meshes. The two spatial discretizations are coupled by a penalty technique at the interface such that the overall semidiscretization satisfies a discrete energy estimate to ensure stability. In addition, optimal convergence is obtained in the sense that when combining a fourth order finite difference method with a discontinuous Galerkin method using third order local polynomials, the overall convergence rate is fourth order. Furthermore, we use a novel approach to derive an error estimate for the semidiscretization by combining the energy method and the normal mode analysis for a corresponding one dimensional model problem. The stability and accuracy analysis are verified in numerical experiments.

翻译：我们根据高度精确的有限差异方法和不连续的Galerkin方法,为第二顺序波形方程式开发了混合空间分解法。混合法结合了笛卡尔电网中有限差异法的计算效率,以及非结构化的网舍中不连续的Galerkin方法的几何灵活性。在结合两种空间分解法的同时,在界面上使用一种惩罚技术,使整个半分化能满足一个离散的能量估计,以确保稳定性。此外,取得最佳汇合,因为当将第四顺序差异法与不连续的Galerkin方法结合时,使用第三顺序本地多式组合法的总趋同率是第四顺序。此外,我们使用一种新颖的方法,将能源方法与相应的一维模型问题的正常模式分析结合起来,得出半分化的误差估计值。在数字实验中验证了稳定性和准确性分析。