Energy plus maximum bound preserving Runge-Kutta methods for the Allen-Cahn equation

TL;DR

This paper develops new Runge-Kutta methods that preserve both the maximum bound and energy dissipation properties for the Allen-Cahn equation, addressing a challenge in high-order numerical scheme design for phase field models.

Contribution

It extends SSP Runge-Kutta theory to nonlinear phase field equations, providing conditions for schemes to preserve maximum bounds and energy dissipation.

Findings

Derived necessary and sufficient conditions for MBP preservation.

Established necessary conditions for energy dissipation law compliance.

Proposed Runge-Kutta schemes satisfying both properties.

Abstract

It is difficult to design high order numerical schemes which could preserve both the maximum bound property (MBP) and energy dissipation law for certain phase field equations. Strong stability preserving (SSP) Runge-Kutta methods have been developed for numerical solution of hyperbolic partial differential equations in the past few decades, where strong stability means the non-increasing of the maximum bound of the underlying solutions. However, existing framework of SSP RK methods can not handle nonlinear stabilities like energy dissipation law. The aim of this work is to extend this SSP theory to deal with the nonlinear phase field equation of the Allen-Cahn type which typically satisfies both maximum bound preserving (MBP) and energy dissipation law. More precisely, for Runge-Kutta time discretizations, we first derive a general necessary and sufficient condition under which MBP is satisfied; and we further provide a necessary condition under which the MBP scheme satisfies energy dissipation.