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

The ubiquity of semilinear parabolic equations is clear from their numerous applications
ranging from physics and biology to materials and social sciences. In this paper, we …

The aim of this work is to develop a conservative, positivity-preserving (PP), nonlinear finite
volume (FV) scheme for the multi-term nonlocal Nagumo-type equations on distorted …

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 …

The scalar auxiliary variable (SAV) approach [31] and its generalized version GSAV
proposed in [20] are very popular methods to construct efficient and accurate energy stable …

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 …

In this paper, we describe a new nonlinear finite-volume scheme that preserves the discrete
maximum principle (DMP) for the two-dimensional sub-diffusion equation on distorted …

In comparison with the Cahn–Hilliard equation, the classic Allen-Cahn equation satisfies the
maximum bound principle (MBP) but fails to conserve the mass along the time. In this paper …