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

The article focuses on the development, stability, and convergence analysis of high-order numerical schemes for one- and two-dimensional nonlocal wave equations posed on unbounded spatial domains. It introduces arbitrarily high-order quadrature-based finite difference schemes for spatial discretization and an explicit finite difference scheme for time discretization, resulting in a fully discrete infinite system. To reduce this infinite system to a finite computational domain, the authors construct discrete Dirichlet-to-Neumann (DtN)-type absorbing boundary conditions (ABCs) based on discrete Dirichlet-to-Dirichlet (DtD)-type mappings and the discrete nonlocal Green's first identity. The DtN-type ABCs enable rigorous stability and convergence proofs under a Courant–Friedrichs–Lewy (CFL) condition, achieving optimal convergence orders of \(\mathcal{O}(\tau^2 + h^q)\), where \(\tau\) and \(h\) are the time step and spatial mesh sizes, respectively. Numerical experiments in both one- and two-dimensional settings validate the theoretical results and demonstrate the effectiveness of the proposed ABCs in preventing artificial wave reflections. The work also discusses challenges related to the computational cost of convolution kernels in the ABCs and outlines directions for future research, including fast algorithms for boundary condition evaluation and higher-order temporal schemes.