We propose a linear, fully decoupled, and energy stable finite difference scheme for solving
a phase-field surfactant fluid system. Inspired by the idea of multiple scalar auxiliary …

In this study, we present the stability analysis of a fully explicit finite difference method (FDM)
for solving the Allen–Cahn (AC) equation. The AC equation is a second-order nonlinear …

Turbulent relaxation is the process of turbulent systems reaching the state of equilibrium,
starting upon quenching the turbulence forcing acting on them. Such relaxation in binary …

In this study, we add nonlocal Lagrange multipliers to the Allen–Cahn (AC) equation to
establish the N-component conservative Allen–Cahn (CAC) system and the ternary …

In this paper, a short review of the calculus of exact finite-differences of integer order is
proposed. The finite-difference operators are called the exact finite-differences of integer …

In this study, we develop first-and second-order time-accurate energy stable methods for the
phase-field crystal equation and the Swift–Hohenberg equation with quadratic-cubic non …

A two-mode phase field crystal (PFC) model is employed to investigate the equilibrium
configurations of a range of grain boundaries in fcc-structured materials. A total of 80 …

In this study, we develop high-order time-accurate, efficient, and energy stable schemes for
solving the conservative Swift–Hohenberg equation that can be used to describe the L 2 …

In this paper, a generalized Allen–Cahn-type phase-field crystal model with face-centered-
cubic ordering structure is presented. By introducing a space-time dependent Lagrange …

Quasicrystals have extensive applications in material sciences. In this article, we develop an
unconditional energy-dissipation-preserving, temporally second-order accurate, and linear …