A Mathematical Analysis of a Smooth-Convex-Concave Splitting Scheme for the Swift--Hohenberg Equation
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908799752732672 |
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| author | Yonekura, Yuki Iwade, Daiki Sato, Shun Matsuo, Takayasu |
| author_facet | Yonekura, Yuki Iwade, Daiki Sato, Shun Matsuo, Takayasu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_22687 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Mathematical Analysis of a Smooth-Convex-Concave Splitting Scheme for the Swift--Hohenberg Equation Yonekura, Yuki Iwade, Daiki Sato, Shun Matsuo, Takayasu Numerical Analysis 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. |
| title | A Mathematical Analysis of a Smooth-Convex-Concave Splitting Scheme for the Swift--Hohenberg Equation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2601.22687 |