On the energy stability of Strang-splitting for Cahn-Hilliard

TL;DR

This paper establishes the first strong energy stability results for second order Strang-splitting methods applied to the Cahn-Hilliard equation, ensuring uniform energy bounds and Sobolev norm persistence.

Contribution

It introduces a new theoretical framework that proves uniform energy stability and higher Sobolev norm persistence for second order operator-splitting methods, resolving longstanding open issues.

Findings

Proves uniform energy stability of the numerical solution.

Shows persistence of all higher Sobolev norms.

First stability result for second order splitting methods for Cahn-Hilliard.

Abstract

We consider a Strang-type second order operator-splitting discretization for the Cahn-Hilliard equation. We introduce a new theoretical framework and prove uniform energy stability of the numerical solution and persistence of all higher Sobolev norms. This is the first strong stability result for second order operator-splitting methods for the Cahn-Hilliard equation. In particular we settle several long-standing open issues in the work of Cheng, Kurganov, Qu and Tang \cite{Tang15}.