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 …

We propose a new numerical technique to deal with nonlinear terms in gradient flows. By
introducing a scalar auxiliary variable (SAV), we construct efficient and robust energy stable …

The scalar auxiliary variable (SAV) method was introduced by Shen et al. in [36] and has
been broadly used to solve thermodynamically consistent PDE problems. By utilizing scalar …

We construct and analyze a class of extrapolated and linearized Runge--Kutta (RK)
methods, which can be of arbitrarily high order, for the time discretization of the Allen--Cahn …

We propose a new Lagrange multiplier approach to design unconditional energy stable
schemes for gradient flows. The new approach leads to unconditionally energy stable …

Numerical schemes to approximate the Cahn–Hilliard equation have been widely studied in
recent times due to its connection with many physically motivated problems. In this work we …

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 …

In this paper, we focus on efficient second-order in time approximations of the Allen–Cahn
and Cahn–Hilliard equations. First of all, we present the equations, generic second-order …

We present a systematic approach to develo** arbitrarily high-order, unconditionally
energy stable numerical schemes for thermodynamically consistent gradient flow models …