Negative time splitting is stable

Abstract

For high order (than two) in time operator-splitting methods applied to dissipative systems, a folklore issue is the appearance of negative-time/backward-in-time linear evolution operators such as backward heat operators interwoven with nonlinear evolutions. The stability of such methods has remained an ensuing difficult open problem. In this work we consider a fourth order operator splitting discretization for the Allen-Cahn equation which is a prototypical high order splitting method with negative time-stepping, i.e. backward in time integration for the linear parabolic part. We introduce a new theoretical framework and prove uniform energy stability and higher Sobolev stability. This is the first strong stability result for negative time stepping operator-splitting methods.

15 pages. arXiv admin note: text overlap with arXiv:2107.05349

References in corpus (4)

• Stability of the semi-implicit method for the Cahn-Hilliard equation with logarithmic potentials
• On the energy stability of Strang-splitting for Cahn-Hilliard
• Stability analysis for the Implicit-Explicit discretization of the Cahn-Hilliard equation
• The operator-splitting method for Cahn-Hilliard is stable