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A novel bond-based nonlocal diffusion model with matrix-valued coefficients in non-divergence form and its collocation discretization

Lili Ju, Hao Tian, Junke Lu

TL;DR

The paper introduces a bond-based nonlocal diffusion operator with matrix-valued coefficients in non-divergence form by embedding the coefficient matrix into a Gaussian-type kernel with covariance δ^2A(x) and applying a truncation B_{δ,A,α}(x) for computational practicality. It establishes well-posedness, a maximum principle, and mass conservation for constant A, and develops a linear collocation discretization that yields an M-matrix system with discrete maximum principle; the method demonstrates exponential convergence on uniform grids and effective δ→0 asymptotic compatibility, validated through comprehensive 2D and 3D experiments for both isotropic and anisotropic diffusion. The approach enables efficient simulation of anisotropic diffusion within a bond-based nonlocal framework and offers clear guidance on truncation and parameter choices (e.g., χ^2_α(d)=36). While the results show strong numerical performance and alignment with local limits, rigorous convergence and compatibility proofs in general dimensions remain an open area for future work.

Abstract

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.

A novel bond-based nonlocal diffusion model with matrix-valued coefficients in non-divergence form and its collocation discretization

TL;DR

The paper introduces a bond-based nonlocal diffusion operator with matrix-valued coefficients in non-divergence form by embedding the coefficient matrix into a Gaussian-type kernel with covariance δ^2A(x) and applying a truncation B_{δ,A,α}(x) for computational practicality. It establishes well-posedness, a maximum principle, and mass conservation for constant A, and develops a linear collocation discretization that yields an M-matrix system with discrete maximum principle; the method demonstrates exponential convergence on uniform grids and effective δ→0 asymptotic compatibility, validated through comprehensive 2D and 3D experiments for both isotropic and anisotropic diffusion. The approach enables efficient simulation of anisotropic diffusion within a bond-based nonlocal framework and offers clear guidance on truncation and parameter choices (e.g., χ^2_α(d)=36). While the results show strong numerical performance and alignment with local limits, rigorous convergence and compatibility proofs in general dimensions remain an open area for future work.

Abstract

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.
Paper Structure (18 sections, 3 theorems, 80 equations, 5 figures, 8 tables)

This paper contains 18 sections, 3 theorems, 80 equations, 5 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Review of existing nonlocal diffusion models
  3. Nonlocal modeling for isotropic diffusion
  4. Nonlocal modeling for anisotropic diffusion
  5. A novel bond-based nonlocal diffusion model with matrix-valued coefficients
  6. A nonlocal operator based on the Gaussian-type kernel function
  7. Convergence to the local diffusion operator in non-divergence form
  8. Truncation of the influence region for the kernel function
  9. Wellposedness and properties
  10. Numerical discretization
  11. A linear collocation discretization scheme
  12. Discussion on convergence and asymptotic compatibility
  13. Numerical experiments
  14. Tests with fixed horizon parameter
  15. Tests for $\delta$-convergence
  16. ...and 3 more sections

Key Result

Theorem 1

(Well-posedness) Let the parameter $\delta>0$ be fixed. Assume that the kernel function $\gamma_{\alpha}({\boldsymbol x},{\boldsymbol y})$ satisfies Then there exists a unique solution $u\in L_{n0}^{2}(\Omega)$ to the nonlocal diffusion problem dbc. Furthermore, the solution satisfies the a priori estimate

Figures (5)

  • Figure 1: Plots of the Gaussian function $p({\boldsymbol x},{\bf 0},\delta^{2} \mathbf{A})$ (in two dimensions) with the isotropic constant coefficient matrix $\mathbf{A}=[1,0; 0,1]$ and $\delta=1/10$ (left) and $\delta=1/20$ (right), respectively. A smaller $\delta$ results in contours that are more elongated and concentrated around the center, while a larger $\delta$ leads to contours that are more spread out and diffuse.
  • Figure 2: When the diffusion coefficient matrix $\mathbf{A} =[1,0;0,1]$, the probability density function is isotropic and thus the projection of its iso-density contours onto the coordinate plane takes the form of a circle (left). When the diffusion coefficient matrix $\mathbf{A}=[10,2;2,1]$, the probability density function is anisotropic and subjected to a combined stretching and rotation, resulting in the rotated image of an elliptical shape upon projection onto the coordinate plane (right).
  • Figure 3: Illustration of the truncated influence region $B_{\delta,{\bf A},\alpha}({\bf 0})$ with $\chi_\alpha^2(2)=36$ (i.e., $\alpha\approx 1.52\times10^{-8}$) in two dimensions. Left: the identity coefficient matrix $\mathbf{A}$=[1,0; 0,1], the truncated influence region is a disk with radius $6\delta$; middle: an anisotropic coefficient matrix $\mathbf{A}$=[10, 0; 0,1], the truncated influence region is an elliptic region whose semi-major axis is $6\sqrt{10} \delta$ and semi-minor axis is $6\delta$; right: $\mathbf{A}=[31/4,-9\sqrt{3}/4;-9\sqrt{3}/4,13/4]$, the truncated influence region of is a rotated (counterclockwisely $30^{\circ}$) elliptic region whose semi-major axis is $6\sqrt{10} \delta$ and semi-minor axis is $6\delta$.
  • Figure 4: Illustration of the computational domain $\Omega=\Omega_s\cup\Omega_c$ and the influence regions with grids in the background.
  • Figure 5: Numerical solutions produced by the linear collocation scheme \ref{['detailed_Interpolate_linear']} for the nonlocal diffusion model \ref{['dbc']} with the four different diffusion coefficient matrices in Example \ref{['expDis']}.

Theorems & Definitions (14)

  • Theorem 1
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Remark 2
  • Example 1
  • Example 2
  • ...and 4 more