Maximum bound principle preserving integrating factor Runge-Kutta methods for semilinear parabolic equations

TL;DR

This paper develops high-order integrating factor Runge-Kutta methods that preserve the maximum bound principle for semilinear parabolic equations, ensuring solutions stay within bounds over time.

Contribution

It introduces the first high-order MBP-preserving IFRK schemes, including a four-stage, fourth-order method applicable to Allen-Cahn equations, with proven error estimates.

Findings

The four-stage, fourth-order IFRK scheme preserves the MBP for certain systems.

Error estimates for the schemes are established and validated.

Numerical simulations demonstrate long-term stability and efficiency.

Abstract

A large class of semilinear parabolic equations satisfy the maximum bound principle (MBP) in the sense that the time-dependent solution preserves for any time a uniform pointwise bound imposed by its initial and boundary conditions. Investigation on numerical schemes of these equations with preservation of the MBP has attracted increasingly attentions in recent years, especially for the temporal discretizations. In this paper, we study high-order MBP-preserving time integration schemes by means of the integrating factor Runge-Kutta (IFRK) method. Beginning with the space-discrete system of semilinear parabolic equations, we present the IFRK method in general form and derive the sufficient conditions for the method to preserve the MBP. In particular, we show that the classic four-stage, fourth-order IFRK scheme is MBP-preserving for some typical semilinear systems although not strong stability preserving, which can be instantly applied to the Allen-Cahn type of equations. In addition, error estimates for these numerical schemes are proved theoretically and verified numerically, as well as their efficiency by simulations of long-time evolutional behavior.