An exponential-free Runge--Kutta framework for developing third-order unconditionally energy stable schemes for the Cahn--Hilliard equation
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866916494619705344 |
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| author | Wang, Haifeng Sun, Jingwei Zhang, Hong Qian, Xu Song, Songhe |
| author_facet | Wang, Haifeng Sun, Jingwei Zhang, Hong Qian, Xu Song, Songhe |
| contents | In this work, we develop a class of up to third-order energy-stable schemes for the Cahn--Hilliard equation. Building on Lawson's integrating factor Runge--Kutta method, which is widely used for stiff semilinear equations, we discuss its limitations, such as the inability to preserve the equilibrium state and the oversmoothing of interfacial layers in the solution's profile because of the exponential damping effects. To overcome this drawback, we approximate the exponential term using a class of sophisticated Taylor polynomials, leading to a novel Runge--Kutta framework called exponential-free Runge--Kutta. By incorporating stabilization techniques, we analyze the energy stability of the proposed schemes and demonstrate that they preserve the original energy dissipation without time-step restrictions. Furthermore, we perform an analysis of the linear stability and establish an error estimate in the $\ell^2$ norm. A series of numerical experiments validate the high-order accuracy, mass conservation, and energy dissipation of our schemes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16271 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An exponential-free Runge--Kutta framework for developing third-order unconditionally energy stable schemes for the Cahn--Hilliard equation Wang, Haifeng Sun, Jingwei Zhang, Hong Qian, Xu Song, Songhe Numerical Analysis 65M12, 65M15, 65M70, 35Q92 G.1.8 In this work, we develop a class of up to third-order energy-stable schemes for the Cahn--Hilliard equation. Building on Lawson's integrating factor Runge--Kutta method, which is widely used for stiff semilinear equations, we discuss its limitations, such as the inability to preserve the equilibrium state and the oversmoothing of interfacial layers in the solution's profile because of the exponential damping effects. To overcome this drawback, we approximate the exponential term using a class of sophisticated Taylor polynomials, leading to a novel Runge--Kutta framework called exponential-free Runge--Kutta. By incorporating stabilization techniques, we analyze the energy stability of the proposed schemes and demonstrate that they preserve the original energy dissipation without time-step restrictions. Furthermore, we perform an analysis of the linear stability and establish an error estimate in the $\ell^2$ norm. A series of numerical experiments validate the high-order accuracy, mass conservation, and energy dissipation of our schemes. |
| title | An exponential-free Runge--Kutta framework for developing third-order unconditionally energy stable schemes for the Cahn--Hilliard equation |
| topic | Numerical Analysis 65M12, 65M15, 65M70, 35Q92 G.1.8 |
| url | https://arxiv.org/abs/2411.16271 |