A spectral collocation method for nonlocal diffusion equations
For researchers in computational mathematics, this provides a more efficient numerical method for nonlocal diffusion models, which are computationally expensive with traditional approaches.
The paper develops a spectral collocation method for nonlocal diffusion equations, achieving exponential convergence rates and significantly reducing computational cost compared to finite difference and finite element methods.
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.