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

In this paper, we consider an exponential scalar auxiliary variable (E-SAV) approach to
obtain energy stable schemes for a class of phase field models. This novel auxiliary variable …

In this paper, we summarize some recent advances related to the energetic variational
approach (EnVarA), a general variational framework of building thermodynamically …

An adaptive BDF2 implicit time-stepping method is analyzed for the phase field crystal
model. The suggested method is proved to preserve a modified energy dissipation law at the …

Liquid crystals are a type of soft matter that is intermediate between crystalline solids and
isotropic fluids. The study of liquid crystals has made tremendous progress over the past four …

The binary fluid surfactant phase-field model, coupled with two Cahn--Hilliard equations and
Navier--Stokes equations, is a very complex nonlinear system, which poses many …

Whether high order temporal integrators can preserve the maximum principle of Allen-Cahn
equation has been an open problem in recent years. This work provides a positive answer …

This paper proposes a novel computational modeling approach to investigate the fluid-
structure interactions with moving contact lines. By embracing the generalized Onsager …

We construct and analyze the variable-step scheme to efficiently solve the space-time
fractional Cahn–Hilliard equation in two dimensions. The associated variational energy …

In recent years, there has been growing interest in developing numerical methods to retain
consistency between the partial differential equation system and its deduced equations (eg …