An Unconditionally Gradient Stable Numerical Method for the Ohta-Kawasaki Model

Abstract

Abstract. We present a finite difference method for solving the OhtaKawasaki model, representing a model of mesoscopic phase separation for the block copolymer. The numerical methods for solving the OhtaKawasaki model need to inherit the mass conservation and energy dissipation properties. We prove these characteristic properties and solvability and unconditionally gradient stability of the scheme by using Hessian matrices of a discrete functional. We present numerical results that validate the mass conservation, and energy dissipation, and unconditional stability of the method.

An adaptive Cahn-Hilliard equation for enhanced edges in binary image inpainting

“…More precisely, we use a semi-implicit approach called convexity splitting (CS) method (see Eyre 14 ) which was originally proposed by Yuille and Rangarajan 15 to solve general optimization problems. It was also applied for different nonlinear models such as the Cahn-Hilliard and Ohta-Kawasaki models (see Bertozzi and Schönlieb 12 ; Kim and Shin 16 ). The idea of CS is to divide the energies (9) and (10) into two parts; a convex plus a concave one.…”

Error estimates for second-order SAV finite element method to phase field crystal model

“…As a well-known approach, the convex splitting approach [8] can guarantee the numerical energy stability. For solving the PFC model, the convex splitting approach combined with different spatial discretization methods is studied, including spectral method [4,19], finite difference method [13,16,29], finite element method [9,26], and local discontinuous Galerkin method [11,12]. However, all second-order schemes in the above study are nonlinear, and their implementations are too complex and costly owing to nonlinear equations to be solved at each time step.…”