A Stable and High-Order Numerical Scheme with Dtn-Type Artificialboundary Conditions for a 2d Peridynamic Diffusion Model

Abstract

In this work, a stable and high-order numerical scheme with DtN-type artificial boundary con-ditions (ABCs) is designed for solving a peridynamic diffusion model on the whole two dimensionaldomain. To do so, we first use a high-order quadrature-based finite difference scheme to approximatethe spatially nonlocal operator, and use the BDF2 scheme to approximate the temporal direction.For the resulting fully discrete system, we apply the nonlocal potential theory to obtain the Dirichlet-to-Dirichlet (DtD)-type ABCs. After that, we further derive the Dirichlet-to-Neumann (DtN)-typeABCs based on the nonlocal Neumann data defined from the discrete nonlocal Green’s first identity.For the stability and convergence analysis, we reformulate the BDF2 operator into a discrete con-volution sum form, then use the technique of the discrete orthogonal convolution kernels to obtainthe optimal convergence order. Finally, numerical experiments are provided to demonstrate theaccuracy and effectiveness of the proposed approach.

Keywords: Peridynamic diffusion model, Unbounded domain, Artificial boundary condition, Stability and convergence, DtN-type map.