An Efficient and Accurate Method for the Conservative Swift–Hohenberg Equation and Its Numerical Implementation
The conservative Swift–Hohenberg equation was introduced to reformulate the phase-field crystal model. A challenge in solving the conservative Swift–Hohenberg equation numerically is how to treat the nonlinear term to preserve mass conservation without compromising efficiency and accuracy. To resolv...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/9/1502 |
| _version_ | 1797554538039738368 |
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| author | Hyun Geun Lee |
| author_facet | Hyun Geun Lee |
| author_sort | Hyun Geun Lee |
| collection | DOAJ |
| description | The conservative Swift–Hohenberg equation was introduced to reformulate the phase-field crystal model. A challenge in solving the conservative Swift–Hohenberg equation numerically is how to treat the nonlinear term to preserve mass conservation without compromising efficiency and accuracy. To resolve this problem, we present a linear, high-order, and mass conservative method by placing the linear and nonlinear terms in the implicit and explicit parts, respectively, and employing the implicit-explicit Runge–Kutta method. We show analytically that the method inherits the mass conservation. Numerical experiments are presented demonstrating the efficiency and accuracy of the proposed method. In particular, long time simulation for pattern formation in 2D is carried out, where the phase diagram can be observed clearly. The MATLAB code for numerical implementation of the proposed method is provided in Appendix. |
| first_indexed | 2024-03-10T16:33:37Z |
| format | Article |
| id | doaj.art-2069beabcc6547a59215e3e8784f7eb4 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T16:33:37Z |
| publishDate | 2020-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-2069beabcc6547a59215e3e8784f7eb42023-11-20T12:35:37ZengMDPI AGMathematics2227-73902020-09-0189150210.3390/math8091502An Efficient and Accurate Method for the Conservative Swift–Hohenberg Equation and Its Numerical ImplementationHyun Geun Lee0Department of Mathematics, Kwangwoon University, Seoul 01897, KoreaThe conservative Swift–Hohenberg equation was introduced to reformulate the phase-field crystal model. A challenge in solving the conservative Swift–Hohenberg equation numerically is how to treat the nonlinear term to preserve mass conservation without compromising efficiency and accuracy. To resolve this problem, we present a linear, high-order, and mass conservative method by placing the linear and nonlinear terms in the implicit and explicit parts, respectively, and employing the implicit-explicit Runge–Kutta method. We show analytically that the method inherits the mass conservation. Numerical experiments are presented demonstrating the efficiency and accuracy of the proposed method. In particular, long time simulation for pattern formation in 2D is carried out, where the phase diagram can be observed clearly. The MATLAB code for numerical implementation of the proposed method is provided in Appendix.https://www.mdpi.com/2227-7390/8/9/1502conservative swift–hohenberg equationlinear methodhigh-order time accuracymass conservationfourier spectral method |
| spellingShingle | Hyun Geun Lee An Efficient and Accurate Method for the Conservative Swift–Hohenberg Equation and Its Numerical Implementation Mathematics conservative swift–hohenberg equation linear method high-order time accuracy mass conservation fourier spectral method |
| title | An Efficient and Accurate Method for the Conservative Swift–Hohenberg Equation and Its Numerical Implementation |
| title_full | An Efficient and Accurate Method for the Conservative Swift–Hohenberg Equation and Its Numerical Implementation |
| title_fullStr | An Efficient and Accurate Method for the Conservative Swift–Hohenberg Equation and Its Numerical Implementation |
| title_full_unstemmed | An Efficient and Accurate Method for the Conservative Swift–Hohenberg Equation and Its Numerical Implementation |
| title_short | An Efficient and Accurate Method for the Conservative Swift–Hohenberg Equation and Its Numerical Implementation |
| title_sort | efficient and accurate method for the conservative swift hohenberg equation and its numerical implementation |
| topic | conservative swift–hohenberg equation linear method high-order time accuracy mass conservation fourier spectral method |
| url | https://www.mdpi.com/2227-7390/8/9/1502 |
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