Stability and error estimates of the SAV Fourier-spectral method for the phase field crystal equation
We consider fully discrete schemes based on the scalar auxiliary variable (SAV) approach and stabilized SAV approach in time and the Fourier-spectral method in space for the phase field crystal (PFC) equation. Unconditionally, energy stability is established for both first- and second-order fully di...
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| Published in | Advances in computational mathematics Vol. 46; no. 3 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
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01.06.2020
Springer Nature B.V |
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| Abstract | We consider fully discrete schemes based on the scalar auxiliary variable (SAV) approach and stabilized SAV approach in time and the Fourier-spectral method in space for the phase field crystal (PFC) equation. Unconditionally, energy stability is established for both first- and second-order fully discrete schemes. In addition to the stability, we also provide a rigorous error estimate which shows that our second-order in time with Fourier-spectral method in space converges with order
O
(Δ
t
2
+
N
−
m
), where Δ
t
,
N
, and
m
are time step size, number of Fourier modes in each direction, and regularity index in space, respectively. We also present numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of the schemes. |
|---|---|
| AbstractList | We consider fully discrete schemes based on the scalar auxiliary variable (SAV) approach and stabilized SAV approach in time and the Fourier-spectral method in space for the phase field crystal (PFC) equation. Unconditionally, energy stability is established for both first- and second-order fully discrete schemes. In addition to the stability, we also provide a rigorous error estimate which shows that our second-order in time with Fourier-spectral method in space converges with order
O
(Δ
t
2
+
N
−
m
), where Δ
t
,
N
, and
m
are time step size, number of Fourier modes in each direction, and regularity index in space, respectively. We also present numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of the schemes. We consider fully discrete schemes based on the scalar auxiliary variable (SAV) approach and stabilized SAV approach in time and the Fourier-spectral method in space for the phase field crystal (PFC) equation. Unconditionally, energy stability is established for both first- and second-order fully discrete schemes. In addition to the stability, we also provide a rigorous error estimate which shows that our second-order in time with Fourier-spectral method in space converges with order O(Δt2 + N−m), where Δt, N, and m are time step size, number of Fourier modes in each direction, and regularity index in space, respectively. We also present numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of the schemes. |
| ArticleNumber | 48 |
| Author | Shen, Jie Li, Xiaoli |
| Author_xml | – sequence: 1 givenname: Xiaoli surname: Li fullname: Li, Xiaoli organization: School of Mathematical Sciences, Xiamen University, Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing, Xiamen University – sequence: 2 givenname: Jie surname: Shen fullname: Shen, Jie email: shen7@purdue.edu organization: Department of Mathematics, Purdue University |
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| Keywords | 65M15 Phase field crystal Error estimates Scalar auxiliary variable (SAV) 65M70 Energy stability Fourier-spectral method 65M12 35G25 |
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| SubjectTerms | Computational mathematics Computational Mathematics and Numerical Analysis Computational Science and Engineering Mathematical and Computational Biology Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Robustness (mathematics) Spectra Spectral methods Stability Visualization |
| Title | Stability and error estimates of the SAV Fourier-spectral method for the phase field crystal equation |
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