SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page A3957-A3978, January 2020.
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 $k$th-order multistep exponential integrator in time and a lumped mass finite element method in space with piecewise $r$th-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 $O(\tau^k+h^r)$. 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.

中文翻译：

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保留抛物线方程最大值原理的任意高阶指数截止方法

SIAM科学计算杂志，第42卷，第6期，第A3957-A3978页，2020年1月。
提出了一类新的高阶最大原理保数值方法，用于求解抛物线方程，并将其应用于半线性Allen-Cahn方程。所提出的方法由时间上的$ k $阶多步指数积分器和具有分段$ r $阶多项式和Gauss-Lobatto正交的空间集中质量有限元方法组成。在每个时间级别，通过切断操作都会在有限元节点处消除违反最大原理的额外值。节点处的剩余值满足最大原理，并证明收敛于$ O（\ tau ^ k + h ^ r）$的误差范围。通过选择大的$ k $和$ r $，可以使精度任意高。