Existing nonlocal diffusion models are predominantly classified into two categories: bond-based models, which involve a single-fold integral and usually simulate isotropic diffusion, and state-based models, which contain a double-fold integral and can additionally prototype anisotropic diffusion. While bond-based models exhibit computational efficiency, they are somewhat limited in their modeling capabilities. In this paper, we develop a novel bond-based nonlocal diffusion model with matrix-valued coefficients in non-divergence form. Our approach incorporates the coefficients into a covariance matrix and employs the multivariate Gaussian function with truncation to define the kernel function, and subsequently model the nonlocal diffusion process through the bond-based formulation. We successfully establish the well-posedness of the proposed model along with deriving some of its properties on maximum principle and mass conservation. Furthermore, an efficient linear collocation scheme is designed for numerical solution of our model. Comprehensive experiments in two and three dimensions are conducted to showcase application of the proposed nonlocal model to both isotropic and anisotropic diffusion problems and to demonstrate numerical accuracy and effective asymptotic compatibility of the proposed collocation scheme.

翻译：现有的非局部扩散模型主要分为两类：涉及单重积分且通常模拟各向同性扩散的基于键的模型，以及包含双重积分并可额外模拟各向异性扩散的基于状态的模型。虽然基于键的模型具有计算效率优势，但其建模能力存在一定局限性。本文提出了一种具有非散度形式矩阵值系数的新型基于键的非局部扩散模型。该方法将系数纳入协方差矩阵，采用截断多元高斯函数定义核函数，进而通过基于键的公式对非局部扩散过程进行建模。我们成功建立了该模型的适定性，并推导了其在最大值原理和质量守恒方面的若干性质。此外，针对模型数值求解设计了高效的线性配置格式。通过二维和三维空间的综合实验，展示了所提非局部模型在各向同性和各向异性扩散问题中的应用，并验证了所提配置格式的数值精度与有效渐近相容性。