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On perfectly matched layers of nonlocal wave equations in unbounded multiscale media

Yu Du and Jiwei Zhang

Vol. 17 (2022), No. 4, 343–364

A nonlocal perfectly matched layer (PML) is formulated for the nonlocal wave equation in the whole real axis, and numerical discretization is designed for solving the reduced PML problem on a bounded domain. The nonlocal PML poses challenges not faced in PDEs. For example, there is no derivative in nonlocal models, which makes it impossible to replace derivatives with complex ones. Here we provide a way of constructing the PML for nonlocal models, which decays the waves exponentially impinging in the layer and makes reflections at the truncated boundary very tiny. To numerically solve the nonlocal PML problem, we design the asymptotically compatible (AC) scheme for a spatially nonlocal operator by combining Talbot’s contour and a Verlet-type scheme for time evolution. The accuracy and effectiveness of our approach are illustrated by various numerical examples.

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nonlocal wave equation, asymptotically compatible (AC) scheme, perfectly matched layer (PML), artificial/absorbing boundary condition (ABC), multiscale media
Received: 7 November 2021
Revised: 16 March 2022
Accepted: 21 March 2022
Published: 18 February 2023
Yu Du
Department of Mathematics
Xiangtan University
Yuhu District, Xiangtan, Hunan
Jiwei Zhang
School of Mathematics and Statistics
Wuhan University
Wuhan, Hubei