| Abstract: |
| In this work we design and analyze first and second order exponential time differencing (ETD) schemes for solving the nonlocal Allen-Cahn (NAC) equation, a generalization of the classic Allen-Cahn equation by replacing the Laplacian with a parameterized nonlocal diffusion operator. The solution of the NAC equation satisfies the maximum principle, thus it is highly desired that the approximated solutions also preserve such a physical property in the fully discrete sense. Our numerical schemes are obtained by using the quadrature-based finite difference method for spatial discretization and applying the stabilized ETD-based approximations for time integration. We prove the discrete maximum principle (DMP) and energy stability of the proposed schemes; in particular, the DMP are unconditionally held for both schemes with respect to any time step size and so does the energy dissipation law for the first-order scheme. We then derive their respective optimal maximum-norm error estimates and further show that the schemes are asymptotic compatible, i.e., the approximated solutions converge to the classic AC solution when the horizon, the spatial mesh size and the time step size all goes to zero. Various experiments are performed to verify these theoretical results, and to numerically investigate the relationship between the discontinuities and the nonlocal parameters. |
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