Exponential function

Classification of discontinuities

Allen–Cahn equation

Mathematics

Algorithm

Mathematical optimization

Laplace operator

Mathematical analysis

Political science

Norm (philosophy)

Parameterized complexity

Maximum principle

Discretization

Applied mathematics

Optimal control

SIAM Journal on Numerical Analysis

The nonlocal Allen--Cahn equation, a generalization of the classic Allen--Cahn equation by replacing the Laplacian with a parameterized nonlocal diffusion operator, satisfies the maximum principle as its local counterpart. In this paper, we develop and analyze first and second order exponential time differencing schemes for solving the nonlocal Allen--Cahn equation, which preserve the discrete maximum principle unconditionally. The fully discrete numerical schemes are obtained by applying the stabilized exponential time differencing approximations for time integration with 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 approximating 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.

Maximum Principle Preserving Exponential Time Differencing Schemes for the Nonlocal Allen--Cahn Equation