Full-rank and low-rank splitting methods for the Swift–Hohenberg equation

Abstract

The Swift–Hohenberg (SH) equation is an important model in the study of pattern formation. In this paper, we propose two full-rank splitting schemes and a low-rank approximation for the SH equation. We first employ a second-order finite difference method to approximate the space derivatives. Based on the resulting semi-discrete system, two full-rank splitting schemes are derived. The convergences of these schemes are studied. For finding a low-rank solution of the SH equation, we derive a low-rank splitting approximation. Numerical results indicate that our methods are robust, accurate and energy dissipation-preserving.