Stabilized integrating factor Runge-Kutta method and unconditional preservation of maximum bound principle

TL;DR

This paper introduces a stabilized integrating factor Runge-Kutta method that unconditionally preserves the maximum bound principle for semilinear parabolic equations, achieving high-order accuracy and demonstrating effectiveness through numerical experiments.

Contribution

It develops a new stabilized IFRK method with proven unconditional MBP preservation, extending high-order schemes and analyzing classical SSP schemes.

Findings

The proposed sIFRK method preserves MBP unconditionally.

Third-order sIFRK schemes are successfully constructed and verified.

Numerical experiments confirm the method's effectiveness.

Abstract

Maximum bound principle (MBP) is an important property for a large class of semilinear parabolic equations, in the sense that the time-dependent solution of the equation with appropriate initial and boundary conditions and nonlinear operator preserves for all time a uniform pointwise bound in absolute value. It has been a challenging problem on how to design unconditionally MBP-preserving high-order accurate time-stepping schemes for these equations. In this paper, we combine the integrating factor Runge-Kutta (IFRK) method with the linear stabilization technique to develop a stabilized IFRK (sIFRK) method, and successfully derive sufficient conditions for the proposed method to preserve MBP unconditionally in the discrete setting. We then elaborate some sIFRK schemes with up to the third-order accuracy, which are proven to be unconditionally MBP-preserving by verifying these conditions. In addition, it is shown that many classic strong stability-preserving sIFRK schemes do not satisfy these conditions except the first-order one. Extensive numerical experiments are also carried out to demonstrate the performance of the proposed method.