Unconditional energy dissipation of Strang splitting for the matrix-valued Allen-Cahn equation

TL;DR

This paper proves that the Strang splitting method unconditionally preserves energy dissipation for the matrix-valued Allen-Cahn equation, ensuring stability, boundedness, and second-order convergence without restrictive time-step constraints.

Contribution

The work removes previous time-step restrictions by providing a refined stability analysis, establishing unconditional energy dissipation and related properties for the Strang splitting method.

Findings

Unconditional energy dissipation is rigorously proven.

The method maintains determinant bounds and stability.

Numerical experiments confirm theoretical results.

Abstract

The energy dissipation property of the Strang splitting method was first demonstrated for the matrix-valued Allen-Cahn (MAC) equation under restrictive time-step constraints [J. Comput. Phys. 454, 110985, 2022]. In this work, we eliminate this limitation through a refined stability analysis framework, rigorously proving that the Strang splitting method preserves the energy dissipation law unconditionally for arbitrary time steps. The refined proof hinges on a precise estimation of the double-well potential term in the modified energy functional. Leveraging this unconditional energy dissipation property, we rigorously establish that the Strang splitting method achieves global-in-time $H^1$-stability, preserves determinant boundedness, and maintains second-order temporal convergence for the matrix-valued Allen-Cahn equation. To validate these theoretical findings, we conduct numerical experiments confirming the method's energy stability and determinant bound preservation for the MAC equation.