Stability and convergence of Strang splitting. Part I: Scalar Allen-Cahn equation

TL;DR

This paper proves stability and energy dissipation properties of second-order Strang splitting methods for Allen-Cahn equations with polynomial and logarithmic nonlinearities, including explicit and implicit schemes.

Contribution

It provides the first rigorous analysis of energy dissipation and stability for Strang splitting methods applied to Allen-Cahn equations with different nonlinearities.

Findings

Unconditional stability for polynomial nonlinearities.

Energy dissipation law established for both polynomial and logarithmic cases.

Maximum principle ensures phase separation in the logarithmic case.

Abstract

We consider a class of second-order Strang splitting methods for Allen-Cahn equations with polynomial or logarithmic nonlinearities. For the polynomial case both the linear and the nonlinear propagators are computed explicitly. We show that this type of Strang splitting scheme is unconditionally stable regardless of the time step. Moreover we establish strict energy dissipation for a judiciously modified energy which coincides with the classical energy up to $\mathcal O(\tau)$ where $\tau$ is the time step. For the logarithmic potential case, since the continuous-time nonlinear propagator no longer enjoys explicit analytic treatments, we employ a second order in time two-stage implicit Runge--Kutta (RK) nonlinear propagator together with an efficient Newton iterative solver. We prove a maximum principle which ensures phase separation and establish energy dissipation law under mild restrictions on the time step. These appear to be the first rigorous results on the energy dissipation of Strang-type splitting methods for Allen-Cahn equations.