Paper Info
Title
Maximum principle preserving exponential time differencing schemes for the nonlocal Allen-Cahn equation
Abstract
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.
Year
DOI
Venue
2019
10.1137/18M118236X
SIAM JOURNAL ON NUMERICAL ANALYSIS
Keywords
Field
DocType
nonlocal Allen-Cahn equation,discrete maximum principle,exponential time differencing,asymptotic compatibility,energy stability
Allen–Cahn equation,Parameterized complexity,Maximum principle,Exponential function,Mathematical analysis,Operator (computer programming),Mathematics,Energy stability,Laplace operator
Journal
Volume
Issue
ISSN
57
2
0036-1429
Citations 
PageRank 
References 
2
0.38
17
Authors
4
Name
Order
Citations
PageRank
Qiang Du11692188.27
Lili Ju244443.43
Xiao Li340.86
Zhonghua Qiao413814.34