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The stability and convergence analysis of high-order numerical approximationsfor the one- and two-dimensional nonlocal wave equations on unbounded spatialdomains are considered. We first use the quadrature-based finite differenceschemes to discretize the spatially nonlocal operator, and apply the explicitdifference scheme to approximate the temporal derivative to achieve a fullydiscrete infinity system. After that, we construct the Dirichlet-to-Neumann(DtN)-type absorbing boundary conditions (ABCs) to reduce the infinite discretesystem 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] toderive the Dirichlet-to-Dirichlet (DtD)-type mappings for one- andtwo-dimensional cases, respectively. We then use the discrete nonlocal Green'sfirst identity to achieve the discrete DtN-type mappings from the DtD-typemappings. The resulting DtN-type mappings make it possible to perform thestability and convergence analysis of the reduced problem. Numericalexperiments are provided to demonstrate the accuracy and effectiveness of theproposed approach.

Stability and convergence analysis of high-order numerical schemes with DtN-type absorbing boundary conditions for nonlocal wave equations