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

We carry out convergence and error analysis of the scalar auxiliary variable (SAV) methods
for L^2 and H^-1 gradient flows with a typical form of free energy. We first derive H^2 …

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 …

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 …

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 …

Understanding the complexity of the nucleation and transition between the crystalline and
quasicrystalline is significant because the structural incommensurability is anisotropic and of …

We present several essential improvements to the powerful scalar auxiliary variable (SAV)
approach. Firstly, by using the introduced scalar variable to control both the nonlinear and …

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