论文标题
保留抛物线方程最大原理的任意高阶指数截止方法
Arbitrarily high-order exponential cut-off methods for preserving maximum principle of parabolic equations
论文作者
论文摘要
提出了一种新的高阶最大原理保存数值方法，用于求解抛物线方程，并应用于半线性allen-cahn方程。所提出的方法包括及时的$ K $ th阶多步指数积分器，以及带有分段$ r $ r $ th-ther-ther-ther-ther ther多项式和高斯 - lobatto quadrature的大量质量有限元方法。在每个时间级别上，违反最大原理的额外值在有限元节点点被截止操作消除。节点点的其余值满足最大原理，并被证明是$ o（τ^k+h^r）$的误差。可以通过选择大型$ k $和$ r $来任意高阶。提供了广泛的数值结果，以说明所提出的方法的准确性以及捕获相位问题模式的有效性。
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 $O(τ^k+h^r)$. 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.
