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 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 Ot2 + Nm), 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.

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The authors express their thanks to the two referees for their helpful suggestions, which led to improvements of the presentation.

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Correspondence to Jie Shen.

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The work of X. Li is supported by the National Natural Science Foundation of China under grant number 11901489, 11971407 and Postdoctoral Science Foundation of China under grant numbers BX20190187 and 2019M650152. The work of J. Shen is supported in part by NSF grants DMS-1620262, DMS-1720442 and AFOSR grant FA9550-16-1-0102.

Communicated by: Carlos Garcia-Cervera

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Li, X., Shen, J. Stability and error estimates of the SAV Fourier-spectral method for the phase field crystal equation. Adv Comput Math 46, 48 (2020).

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  • Phase field crystal
  • Fourier-spectral method
  • Scalar auxiliary variable (SAV)
  • Energy stability
  • Error estimates

Mathematics Subject Classification (2010)

  • 35G25
  • 65M12
  • 65M15
  • 65M70