This chapter surveys recent numerical advances in the phase field method for geometric
surface evolution and related geometric nonlinear partial differential equations (PDEs) …

We propose a new approach, which we term as scalar auxiliary variable (SAV) approach, to
construct efficient and accurate time discretization schemes for a large class of gradient …

In this work, we investigate the two-step backward differentiation formula (BDF2) with
nonuniform grids for the Allen--Cahn equation. We show that the nonuniform BDF2 scheme …

Gradient flow models attract much attention these years. The energy naturally decreases
along the direction of gradient flows. In order to preserve this property, various numerical …

We present and analyze a second order in time variable step BDF2 numerical scheme for
the Cahn--Hilliard equation. The construction relies on a second order backward difference …

We present a second order energy stable numerical scheme for the two and three
dimensional Cahn–Hilliard equation, with Fourier pseudo-spectral approximation in space …

It is known that the Allen-Chan equations satisfy the maximum principle. Is this true for
numerical schemes? To the best of our knowledge, the state-of-art stability framework is the …

This paper is concerned with the generalized Allen–Cahn equation with a nonlinear mobility
that may be degenerate, which also includes an advection term appearing in many …

We discuss how to combine exponential time differencing technique with multi-step method
to develop higher order in time linear numerical scheme that are energy stable for certain …

In this paper, we consider the mathematical modeling and numerical approximation for the
fluid-surfactant phase field model coupled with the Navier–Stokes equations on surfaces …