Generalized SAV-exponential integrator schemes for Allen--Cahn type gradient flows

Abstract

The energy dissipation law and the maximum bound principle (MBP) are two important physical features of the well-known Allen--Cahn equation. While some commonly used first-order time stepping schemes have turned out to preserve unconditionally both the energy dissipation law and the MBP for the equation, restrictions on the time step size are still needed for existing second-order or even higher order schemes in order to have such simultaneous preservation. In this paper, we develop and analyze novel first- and second-order linear numerical schemes for a class of Allen--Cahn type gradient flows. Our schemes combine the generalized scalar auxiliary variable (SAV) approach and the exponential time integrator with a stabilization term, while the standard central difference stencil is used for discretization of the spatial differential operator. We not only prove their unconditional preservation of the energy dissipation law and the MBP in the discrete setting, but we also derive their optimal temporal error estimates under fixed spatial mesh. Numerical experiments are also carried out to demonstrate the properties and performance of the proposed schemes.