  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.
