Abstract The conservative Allen–Cahn equation with a nonlocal Lagrange multiplier satisfies
mass conservation and energy dissipation property. A challenge to numerically solving the …

We consider numerical approximations for the modified phase field crystal equation in this
paper. The model is a nonlinear damped wave equation that includes both diffusive …

In this paper, we present a second-order accurate and linear numerical scheme for the
phase field crystal equation and prove its convergence in the discrete sense. The key …

We present second order, fully discrete, energy stable methods on spatially staggered grids
for a hydrodynamic phase field model of binary viscous fluid mixtures in a confined geometry …

We present temporally first-and second-order accurate methods for the Swift–Hohenberg
(SH) equation with quadratic–cubic nonlinearity. In order to handle the nonconvex …

Abstract The Cahn-Hilliard-Navier-Stokes (CHNS) equations represent the fundamental
building blocks of hydrodynamic phase-field models for multiphase fluid flow dynamics. Due …

In this paper, we extend the phase field crystal model to the modified phase field crystal
model which includes diffusive dynamics, elastic interaction, and stochastic noises effect …

Abstract The Swift–Hohenberg (SH) energy functional has been widely used to study pattern
formation. The L 2-and H− 1-gradient flows for the SH energy functional are the SH and …

We design first-order and second-order time-stepping schemes for the modified phase field
crystal model based on the scalar auxiliary variable method in this work. The model is a …

We carry out error estimates for a linear, second-order, unconditionally energy stable, semi-
discrete time stepping scheme, which is based on the Lagrange Multiplier approach and …