A Second Order Energy Stable BDF Numerical Scheme for the Swift-Hohenberg Equation

In this paper, we propose and analyze a second-order energy stable numerical scheme for the Swift-Hohenberg equation, with a mixed finite element approximation in space. We employ second-order backward differentiation formula scheme with a second-order stabilized term, which guarantees the long time energy stability. We prove that our two-step scheme is unconditionally energy stable and uniquely solvable. Furthermore, we present an optimal error estimate for the scheme. In the end, several numerical experiments are presented to support our theoretical analysis.

Automated summary

This paper proposes and analyzes a second-order energy stable numerical scheme for the Swift-Hohenberg equation, utilizing mixed finite element approximation in space. The scheme employs a second-order backward differentiation formula scheme with a second-order stabilized term to guarantee long-term energy stability. It is proven to be unconditionally energy stable and uniquely solvable, with an optimal error estimate provided.Numerical experiments are conducted to support the theoretical analysis.