A Mathematical Analysis of a Smooth-Convex-Concave Splitting Scheme for the Swift--Hohenberg Equation

The Swift–Hohenberg equation is a widely studied fourth-order model, originally proposed to describe hydrodynamic fluctuations. It admits an energy-dissipation law and, under suitable assumptions, bounded solutions. Many structure-preserving numerical schemes have been proposed to retain such properties; however, existing approaches are often fully implicit and therefore computationally expensive. We introduce a simple design principle for constructing dissipation-preserving finite difference schemes and apply it to the Swift–Hohenberg equation in three spatial dimensions. Our analysis relies on discrete inequalities for the underlying energy, assuming a Lipschitz continuous gradient and either convexity or $μ$-strong convexity of the relevant terms. The resulting method is linearly implicit, yet it preserves the original energy-dissipation law, guarantees unique solvability, ensures boundedness of numerical solutions, and admits an a priori error estimate, provided that the time step is sufficiently small. To the best of our knowledge, this is the first linearly implicit finite difference scheme for the Swift–Hohenberg equation for which all of these properties are established.

💡 Research Summary

The paper addresses the numerical solution of the Swift–Hohenberg equation, a fourth‑order nonlinear PDE that models hydrodynamic fluctuations and possesses a natural energy‑dissipation law. While many structure‑preserving schemes have been proposed, most are fully implicit and therefore computationally expensive because they require solving a nonlinear system at every time step. The authors introduce a systematic design principle called the “Smooth‑Convex‑Concave Splitting Scheme” and apply it to the three‑dimensional Swift–Hohenberg equation, obtaining a linearly implicit finite‑difference method that retains all the desirable analytical properties of the continuous problem.

The continuous equation is written as
∂ₜu = −{∇ ν₁(u)+∇ ν₂(u)−∇ ν₃(u)}
where ν₁, ν₂, ν₃ are scalar functionals of the solution. The key idea is to split the right‑hand side into three parts with distinct convexity properties: ν₁ is assumed L‑smooth (its gradient is Lipschitz), ν₂ is µ₂‑strongly convex, and ν₃ is µ₃‑strongly convex (possibly with µ₃ ≥ 0). The time discretisation reads

Uⁿ⁺¹−Uⁿ = −Δt