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

A spectral method for nonlocal diffusion operators on the sphere