A spectral method for nonlocal diffusion operators on the sphere

TL;DR

This paper introduces a spectral method leveraging spherical harmonics for solving nonlocal diffusion equations on the sphere with high accuracy, enabling efficient simulations of models like Poisson, Allen--Cahn, and Brusselator.

Contribution

The paper develops a spectral algorithm that diagonalizes nonlocal diffusion operators on the sphere, achieving high accuracy and efficiency for time-dependent models.

Findings

Spectral accuracy achieved in solving nonlocal diffusion models.

Efficient eigenvalue computation using quadrature and asymptotic formulas.

Successful application to Poisson, Allen--Cahn, and Brusselator equations.

Abstract

We present algorithms for solving spatially nonlocal diffusion models on the unit sphere with spectral accuracy in space. Our algorithms are based on the diagonalizability of nonlocal diffusion operators in the basis of spherical harmonics, the computation of their eigenvalues to high relative accuracy using quadrature and asymptotic formulas, and a fast spherical harmonic transform. These techniques also lead to an efficient implementation of high-order exponential integrators for time-dependent models. We apply our method to the nonlocal Poisson, Allen--Cahn and Brusselator equations.