Arbitrarily high-order exponential cut-off methods for preserving maximum principle of parabolic equations

TL;DR

This paper introduces a high-order numerical method combining exponential integrators and finite element techniques to solve parabolic equations while preserving the maximum principle, with proven convergence and demonstrated effectiveness.

Contribution

A novel high-order maximum principle preserving method for parabolic equations using exponential integrators and finite elements with a cut-off operation.

Findings

Method achieves arbitrarily high-order accuracy.

Numerical results confirm effectiveness in phase-field pattern capturing.

Error bounds of O(τ^k + h^r) are established.

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 $k$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.