Long-time stability analysis of an explicit exponential Runge-Kutta scheme for Cahn-Hilliard equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915656505491456 |
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| author | Guo, Jing |
| author_facet | Guo, Jing |
| contents | In this paper, we present a comprehensive long-time stability analysis of a second-order explicit exponential Runge--Kutta (ERK2) method for the Cahn--Hilliard (CH) equation. By employing Fourier spectral collocation in space and a two-stage ERK2 scheme in time, we construct a fully discrete numerical method that preserves the original energy dissipation property. The uniform-in-time boundedness of the numerical solution is rigorously proven in the discrete $H^1$ and $H^2$ norms under a mild time-step condition, and an $\ell^\infty$ bound is derived via a discrete Sobolev embedding. These results remove the typical boundedness assumption required in previous energy-stability analyses, thereby establishing unconditional energy dissipation for the fully discrete scheme. Building on this uniform boundedness, we derive an optimal-order error estimate in the $\ell^2$ norm. The analytical framework developed herein is general and can be extended to higher-order exponential integrators for a broader class of phase-field models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05608 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Long-time stability analysis of an explicit exponential Runge-Kutta scheme for Cahn-Hilliard equations Guo, Jing Numerical Analysis 35K58, 65M12, 65M15, 65M70 In this paper, we present a comprehensive long-time stability analysis of a second-order explicit exponential Runge--Kutta (ERK2) method for the Cahn--Hilliard (CH) equation. By employing Fourier spectral collocation in space and a two-stage ERK2 scheme in time, we construct a fully discrete numerical method that preserves the original energy dissipation property. The uniform-in-time boundedness of the numerical solution is rigorously proven in the discrete $H^1$ and $H^2$ norms under a mild time-step condition, and an $\ell^\infty$ bound is derived via a discrete Sobolev embedding. These results remove the typical boundedness assumption required in previous energy-stability analyses, thereby establishing unconditional energy dissipation for the fully discrete scheme. Building on this uniform boundedness, we derive an optimal-order error estimate in the $\ell^2$ norm. The analytical framework developed herein is general and can be extended to higher-order exponential integrators for a broader class of phase-field models. |
| title | Long-time stability analysis of an explicit exponential Runge-Kutta scheme for Cahn-Hilliard equations |
| topic | Numerical Analysis 35K58, 65M12, 65M15, 65M70 |
| url | https://arxiv.org/abs/2512.05608 |