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

TL;DR

This paper introduces a novel linearly implicit finite difference scheme for the Swift--Hohenberg equation that preserves energy dissipation, guarantees bounded solutions, and provides an error estimate, improving computational efficiency while maintaining key properties.

Contribution

The paper presents the first linearly implicit finite difference scheme for the Swift--Hohenberg equation that ensures energy dissipation, boundedness, and solvability with an error estimate.

Findings

Scheme preserves energy-dissipation law.

Guarantees unique solvability and bounded solutions.

Provides an a priori error estimate.

Abstract

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 $\mu$-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.