Stability and error estimates of the SAV Fourier-spectral method for the phase field crystal equation

• Published in: Advances in computational mathematics Vol. 46; no. 3
• Main Authors: Li, Xiaoli, Shen, Jie
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2020; Springer Nature B.V
• Subjects: 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; 65M15; Phase field crystal; Error estimates; Scalar auxiliary variable (SAV); 65M70; Energy stability; Fourier-spectral method; 65M12; 35G25
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-020-09789-9

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.