# Low regularity integrators for semilinear parabolic equations with
maximum bound principles

Cao-Kha Doan, Thi-Thao-Phuong Hoang, Lili Ju, Katharina Schratz

Submitted on 7 November 2022

## Abstract

This paper is concerned with conditionally structure-preserving, low
regularity time integration methods for a class of semilinear parabolic
equations of Allen-Cahn type. Important properties of such equations include
maximum bound principle (MBP) and energy dissipation law; for the former, that
means the absolute value of the solution is pointwisely bounded for all the
time by some constant imposed by appropriate initial and boundary conditions.
The model equation is first discretized in space by the central finite
difference, then by iteratively using Duhamel's formula, first- and
second-order low regularity integrators (LRIs) are constructed for time
discretization of the semi-discrete system. The proposed LRI schemes are proved
to preserve the MBP and the energy stability in the discrete sense.
Furthermore, their temporal error estimates are also successfully derived under
a low regularity requirement that the exact solution of the semi-discrete
problem is only assumed to be continuous in time. Numerical results show that
the proposed LRI schemes are more accurate and have better convergence rates
than classic exponential time differencing schemes, especially when the
interfacial parameter approaches zero.