Maximum bound principle and original energy dissipation of arbitrarily high-order rescaled exponential time differencing Runge-Kutta schemes for Allen--Cahn equations

TL;DR

This paper introduces high-order exponential time differencing Runge-Kutta schemes for Allen--Cahn equations that preserve the original energy dissipation law and maximum bound principle, ensuring physical fidelity and high accuracy.

Contribution

It develops arbitrarily high-order ETDRK schemes that unconditionally preserve the original energy dissipation and maximum bound principle for Allen--Cahn equations.

Findings

Schemes preserve original energy dissipation law.

Schemes guarantee maximum bound principle.

Numerical experiments confirm theoretical results.

Abstract

The energy dissipation law and the maximum bound principle are two critical physical properties of the Allen--Cahn equations. While many existing time-stepping methods are known to preserve the energy dissipation law, most apply to a modified form of energy. In this work, we demonstrate that, when the nonlinear term of the Allen--Cahn equation is Lipschitz continuous, a class of arbitrarily high-order exponential time differencing Runge--Kutta (ETDRK) schemes preserve the original energy dissipation property, under a mild step-size constraint. Additionally, we guarantee the Lipschitz condition on the nonlinear term by applying a rescaling post-processing technique, which ensures that the numerical solution unconditionally satisfies the maximum bound principle. Consequently, our proposed schemes maintain both the original energy dissipation law and the maximum bound principle and can achieve arbitrarily high-order accuracy. We also establish an optimal error estimate for the proposed schemes. Some numerical experiments are carried out to verify our theoretical results.