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 …

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 nonlocal Allen--Cahn equation, a generalization of the classic Allen--Cahn equation by
replacing the Laplacian with a parameterized nonlocal diffusion operator, satisfies the …

Abstract The Molecular Beam Epitaxial model is derived from the variation of a free energy,
that consists of either a fourth order Ginzburg–Landau double well potential or a nonlinear …

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

For the time-fractional phase-field models, the corresponding energy dissipation law has not
been well studied on both the continuous and the discrete levels. In this work, we address …

The energy dissipation law and the maximum bound principle (MBP) are two important
physical features of the well-known Allen--Cahn equation. While some commonly used first …

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 …

The maximum bound principle (MBP) is an important property for a large class of semilinear
parabolic equations, in the sense that the time-dependent solution of the equation with …