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

In this paper we present and analyze finite difference numerical schemes for the Cahn-
Hilliard equation with a logarithmic Flory Huggins energy potential. Both first and second …

In this paper we propose and analyze a second order accurate (in time) numerical scheme
for the square phase field crystal equation, a gradient flow modeling crystal dynamics at the …

In this paper we propose and analyze an energy stable numerical scheme for the Cahn–
Hilliard equation, with second order accuracy in time and the fourth order finite difference …

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 …

In this paper, we construct and analyze a uniquely solvable, positivity preserving and
unconditionally energy stable finite-difference scheme for the periodic three-component …

In this paper, we propose and analyze a positivity-preserving, energy stable numerical
scheme for a certain type of reaction-diffusion systems involving the Law of Mass Action with …

In this paper we propose and analyze an energy stable numerical scheme for the square
phase field crystal (SPFC) equation, a gradient flow modeling crystal dynamics at the atomic …

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 …

We present and analyze a second order in time variable step BDF2 numerical scheme for
the Cahn--Hilliard equation. The construction relies on a second order backward difference …