A new conservative Swift–Hohenberg equation and its mass conservative method

Abstract

The Swift–Hohenberg (SH) energy functional has been widely used to study pattern formation. The L2- and H−1-gradient flows for the SH energy functional are the SH and phase-field crystal (PFC) equations, respectively. The SH equation is of lower-order in space than the PFC equation but does not conserve the total mass. Furthermore, when the SH energy functional does not have the cubic nonlinearity, the SH equation, unlike the PFC equation, only shows striped patterns even for various parameter values. In this study, we introduce a new mass conservative SH equation by using a Lagrange multiplier. In order to solve the conservative SH equation that is an integro–partial differential equation, we propose operator splitting methods that are shown analytically to inherit the mass conservation. Numerical examples including standard tests in the PFC equation are presented to show the applicability of the proposed framework.