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Title: Arbitrarily high-order exponential cut-off methods for preserving maximum principle of parabolic equations
Authors: Li, B 
Yang, J
Zhou, Z 
Issue Date: 2020
Source: SIAM journal on scientific computing, 2020, v. 42, no. 6, p. A3957-A3968
Abstract: A new class of high-order maximum principle preserving numerical methods is proposed for solving parabolic equations, with application to the semilinear Allen-Cahn equation. The proposed method consists of a kth-order multistep exponential integrator in time and a lumped mass finite element method in space with piecewise rth-order polynomials and Gauss-Lobatto quadrature. At every time level, the extra values violating the maximum principle are eliminated at the finite element nodal points by a cut-off operation. The remaining values at the nodal points satisfy the maximum principle and are proved to be convergent with an error bound of O(τ k + hr). The accuracy can be made arbitrarily high-order by choosing large k and r. Extensive numerical results are provided to illustrate the accuracy of the proposed method and the effectiveness in capturing the pattern of phase-field problems.
Keywords: Allen-Cahn equation
Exponential integrator
High order
Lumped mass
Maximum principle
Parabolic equation
Publisher: Society for Industrial and Applied Mathematics
Journal: SIAM journal on scientific computing 
ISSN: 1064-8275
EISSN: 1095-7197
DOI: 10.1137/20M1333456
Rights: © 2020, Society for Industrial and Applied Mathematics.
Unauthorized reproduction of this article is prohibited.
First Published in SIAM Journal on SIAM Journal on Scientific Computing in Volume 42, Issue 6, published by the Society for Industrial and Applied Mathematics (SIAM)
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