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

Yuki Yonekura, Daiki Iwade, Shun Sato, Takayasu Matsuo

TL;DR

The paper develops a first-linearly implicit, dissipation-preserving finite-difference scheme for the 3D Swift–Hohenberg equation by a smooth-convex-concave splitting of the energy. It proves unique solvability, a discrete energy-dissipation law, boundedness of numerical solutions, and an a priori error estimate under a time-step restriction, providing a computationally efficient alternative to fully implicit methods. The approach is anchored in discrete convex analysis and a careful handling of energy components, yielding stability and convergence with explicit treatment of nonlinear terms. The results offer a practical, theoretically grounded framework for structure-preserving simulation of the Swift–Hohenberg dynamics in three dimensions.

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 $μ$-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.

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

TL;DR

The paper develops a first-linearly implicit, dissipation-preserving finite-difference scheme for the 3D Swift–Hohenberg equation by a smooth-convex-concave splitting of the energy. It proves unique solvability, a discrete energy-dissipation law, boundedness of numerical solutions, and an a priori error estimate under a time-step restriction, providing a computationally efficient alternative to fully implicit methods. The approach is anchored in discrete convex analysis and a careful handling of energy components, yielding stability and convergence with explicit treatment of nonlinear terms. The results offer a practical, theoretically grounded framework for structure-preserving simulation of the Swift–Hohenberg dynamics in three dimensions.

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 -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.
Paper Structure (16 sections, 19 theorems, 153 equations)

This paper contains 16 sections, 19 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions of the Computational Domain and Difference Operators
  4. Discrete Function Space and Convex Analysis
  5. Notation for Solutions
  6. Dissipation-Preserving Scheme Framework
  7. Definition of a Smooth-Convex-Concave Splitting Scheme
  8. Error Analysis for the General Smooth-Convex-Concave Splitting Scheme
  9. A Smooth-Convex-Concave Splitting Scheme for the Swift--Hohenberg Equation
  10. Properties of the Swift--Hohenberg equation
  11. Proposed Numerical Scheme
  12. Stability and Dissipation Properties
  13. Error Analysis for the Proposed Scheme
  14. Asymptotic behavior
  15. Proof of \ref{['equ:u_bound']}
  16. ...and 1 more sections

Key Result

Lemma 2.1

Furihata For any three-dimensional grid functions ${f_{{i},{j},{k}}}$, and ${g_{{i},{j},{k}}}$, if then where

Theorems & Definitions (40)

  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 3.1
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Theorem 3.2
  • proof
  • ...and 30 more