A spectral collocation method for nonlocal diffusion equations

TL;DR

This paper introduces a spectral collocation method for nonlocal diffusion equations, offering high-order convergence and reduced computational costs compared to traditional methods, with theoretical error analysis and numerical validation.

Contribution

The paper develops a spectral collocation approach for nonlocal diffusion equations, providing rigorous error analysis and demonstrating exponential convergence.

Findings

Spectral collocation method achieves exponential convergence.

Numerical results confirm high accuracy and efficiency.

Method reduces computational cost compared to finite difference and finite element methods.

Abstract

Nonlocal diffusion model provides an appropriate description of the diffusion process of solute in the complex medium, which cannot be described properly by classical theory of PDE. However, the operators in the nonlocal diffusion models are nonlocal, so the resulting numerical methods generate dense or full stiffness matrices. This imposes significant computational and memory challenge for a nonlocal diffusion model. In this paper, we develop a spectral collocation method for the nonlocal diffusion model and provide a rigorous error analysis which theoretically justifies the spectral rate of convergence provided that the kernel functions and the source functions are sufficiently smooth. Compared to finite difference methods and finite element methods, because of the high order convergence rates, the numerical cost of spectral collocation methods will be greatly decreased. Numerical results confirm the exponential rate of convergence.