Maximum principle preserving exponential time differencing schemes for the nonlocal Allen-Cahn equation
This work provides provably stable and accurate numerical methods for the nonlocal Allen-Cahn equation, which is important for modeling phase transitions with nonlocal interactions.
The authors developed first and second order exponential time differencing schemes for the nonlocal Allen-Cahn equation that unconditionally preserve the discrete maximum principle, with optimal error estimates and asymptotic compatibility proven. Numerical experiments verified the theoretical results.
The nonlocal Allen-Cahn (NAC) equation is a generalization of the classic Allen-Cahn equation by replacing the Laplacian with a parameterized nonlocal diffusion operator, and satisfies the maximum principle as its local counterpart. In this paper, we develop and analyze first and second order exponential time differencing (ETD) schemes for solving the NAC equation, which unconditionally preserve the discrete maximum principle. The fully discrete numerical schemes are obtained by applying the stabilized ETD approximations for time integration with the quadrature-based finite difference discretization in space. We derive their respective optimal maximum-norm error estimates and further show that the proposed schemes are asymptotically compatible, i.e., the approximate solutions always converge to the classic Allen-Cahn solution when the horizon, the spatial mesh size and the time step size go to zero. We also prove that the schemes are energy stable in the discrete sense. Various experiments are performed to verify these theoretical results and to investigate numerically the relation between the discontinuities and the nonlocal parameters.