Asymptotically Compatible Fourier Spectral Approximations of Nonlocal Allen-Cahn Equations
We study Fourier spectral approximations of a nonlocal Allen--Cahn (NAC) equation that reduces to the conventional Allen--Cahn equation in the local limit. We show that the Fourier spectral methods are asymptotically compatible in the sense that they provide convergent approximations to both nonlocal and local models. Furthermore, we provide various error estimates. In particular, it is shown that the numerical solutions of nonlocal models converge to those of the corresponding local models uniformly at a rate of $O(\delta^2)$. This is achieved by first establishing a similar result for linear nonlocal diffusion equations. A careful investigation, both analytically and computationally, is made of the steady state solutions of NAC equations, demonstrating how discontinuities may appear in solutions and how they are related to model parameters.