论文标题
具有DTN型的高阶数值方案的稳定性和收敛分析吸收了非局部波方程的边界条件
Stability and convergence analysis of high-order numerical schemes with DtN-type absorbing boundary conditions for nonlocal wave equations
论文作者
论文摘要
考虑了在无限的空间域上对一维非局部波方程的高阶数值近似值的稳定性和收敛分析。我们首先使用基于正交的有限差方案来离散空间非局部运算符，并应用显式差异方案以近似时间导数以实现完全离散的无限系统。之后，我们将吸收边界条件（ABC）构造为dirichlet到neumann（DTN）型，以将无限的离散系统降低到有限的离散系统中。为此，我们首先在[DU，Zhang和Zheng，\ emph {commun。计算。 Phys。}，24（4）：1049--1072，2018和DU，Han，Zhang和Zheng，\ Emph {Siam J. Sci。 comp。}，40（3）：A1430-A1445，2018]分别针对一维情况和二维案例得出Dirichlet到Dirichlet（DTD）型映射。然后，我们使用离散的非局部绿色的第一个身份来获得DTD型映射的离散DTN型映射。所得的DTN型映射使对减少问题的稳定性和收敛分析成为可能。提供了数值实验来证明所提出方法的准确性和有效性。
The stability and convergence analysis of high-order numerical approximations for the one- and two-dimensional nonlocal wave equations on unbounded spatial domains are considered. We first use the quadrature-based finite difference schemes to discretize the spatially nonlocal operator, and apply the explicit difference scheme to approximate the temporal derivative to achieve a fully discrete infinity system. After that, we construct the Dirichlet-to-Neumann (DtN)-type absorbing boundary conditions (ABCs) to reduce the infinite discrete system into a finite discrete system. To do so, we first adopt the idea in [Du, Zhang and Zheng, \emph{Commun. Comput. Phys.}, 24(4):1049--1072, 2018 and Du, Han, Zhang and Zheng, \emph{SIAM J. Sci. Comp.}, 40(3):A1430--A1445, 2018] to derive the Dirichlet-to-Dirichlet (DtD)-type mappings for one- and two-dimensional cases, respectively. We then use the discrete nonlocal Green's first identity to achieve the discrete DtN-type mappings from the DtD-type mappings. The resulting DtN-type mappings make it possible to perform the stability and convergence analysis of the reduced problem. Numerical experiments are provided to demonstrate the accuracy and effectiveness of the proposed approach.
