We develop a stable finite difference method for the elastic wave equation in bounded media, where the material properties can be discontinuous at curved interfaces. The governing equation is discretized in second order form by a fourth or sixth order accurate summation-by-parts operator. The mesh size is determined by the velocity structure of the material, resulting in nonconforming grid interfaces with hanging nodes. We use order-preserving interpolation and the ghost point technique to couple adjacent mesh blocks in an energy-conserving manner, which is supported by a fully discrete stability analysis. In our previous work for the wave equation, two pairs of order-preserving interpolation operators are needed when imposing the interface conditions weakly by a penalty technique. Here, we only use one pair in the ghost point method. In numerical experiments, we demonstrate that the convergence rate is optimal, and is the same as when a globally uniform mesh is used in a single domain. In addition, with a predictor-corrector time integration method, we obtain time stepping stability with stepsize almost the same as given by the usual Courant-Friedrichs-Lewy condition.
翻译:我们为封闭介质中的弹性波方程式开发一个稳定的有限差异法, 使材料特性在曲线界面中不连续。 调节方程式由第四或第六顺序准确的对数操作员以第二顺序形式分解。 网状尺寸由材料的速度结构决定, 导致与挂点的网格接口不兼容。 我们使用命令保护插座和幽点技术, 以节能的方式将相邻网状区块对齐, 并得到完全离散的稳定分析的支持。 在我们以前对波方公式的工作中, 需要两对保持秩序的内插操作员在用惩罚技术将界面条件强制弱化时使用两对对。 在这里, 我们只使用鬼点法中的一对。 在数字实验中, 我们证明趋同率是最佳的, 与在一个域使用全球统一的网块时一样。 此外, 我们用预测器- 时间整合法, 我们得到了时间步稳定, 与通常的Colant- Fried-L 条件一样, 步步步态几乎相同 。