[Paper Review] Stability and Error estimates of the SAV Fourier-spectral method for the Phase Field Crystal Equation

Xiaoli Li, Jie Shen|arXiv (Cornell University)|Jul 17, 2019

Solidification and crystal growth phenomena16 references4 citations

TL;DR

This paper proposes a stabilized scalar auxiliary variable (SAV) Fourier-spectral method for the phase field crystal (PFC) equation, combining the SAV approach in time with spectral spatial discretization to achieve unconditionally energy-stable, linear schemes. It provides the first rigorous optimal error estimate for fully discrete linear schemes, showing second-order convergence in time and spectral accuracy in space, with stabilization crucial for accuracy at large time steps despite not being required for stability.

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 $O(Δt^2+N^{-m})$, 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.

Motivation & Objective

• Develop unconditionally energy-stable, linear numerical schemes for the phase field crystal (PFC) equation to overcome the complexity of nonlinear schemes.
• Address the lack of convergence analysis and error estimates in existing PFC numerical methods, especially for linear schemes.
• Establish rigorous error estimates for fully discrete schemes using the SAV framework and spectral spatial discretization.
• Investigate the role of stabilization in maintaining accuracy at large time steps without compromising stability.
• Demonstrate the robustness and accuracy of the proposed schemes through numerical experiments.

Proposed method

• Apply the scalar auxiliary variable (SAV) approach to reformulate the PFC equation into a linear system, enabling energy stability and simplifying time discretization.
• Use the Fourier-spectral method for spatial discretization to achieve spectral accuracy in space, leveraging periodic boundary conditions.
• Introduce a stabilized SAV (S-SAV) formulation by adding a stabilization term to improve accuracy at large time steps.
• Construct fully discrete schemes using Crank-Nicolson or backward differentiation formulas in time, ensuring unconditional energy stability.
• Derive uniform bounds on discrete solutions using energy stability, which enables rigorous error analysis.
• Perform error analysis based on the discrete energy stability and regularity assumptions, leading to optimal convergence rates.

Experimental results

Research questions

• RQ1 Can the SAV approach be extended to the PFC equation to yield linear, unconditionally energy-stable schemes with optimal convergence rates?
• RQ2 What is the role of the stabilization term in the S-SAV formulation for achieving accurate results at large time steps?
• RQ3 Is it possible to derive rigorous error estimates for fully discrete linear schemes applied to the PFC equation, given the high-order and nonlinear nature of the problem?
• RQ4 How does the choice of time step size affect the accuracy of the SAV-Fourier-spectral scheme, especially when stabilization is or is not applied?
• RQ5 To what extent does the spectral spatial discretization contribute to the overall convergence rate and numerical efficiency of the scheme?

Key findings

• The proposed second-order SAV-Fourier-spectral scheme achieves optimal convergence rates of $ O( au^2 + N^{-m}) $ in time and space, respectively, where $ au $ is the time step and $ N $ is the number of Fourier modes.
• Unconditional energy stability is proven for both first- and second-order fully discrete schemes, ensuring robustness regardless of time step size.
• The stabilization term in the S-SAV formulation is not required for stability or convergence but is essential for achieving reasonable accuracy at large time steps.
• Numerical experiments confirm second-order convergence in time and spectral convergence in space, validating the theoretical error estimates.
• The schemes accurately capture complex crystal growth dynamics, including dendritic patterns, dislocations, and interface motion, even with large time steps when stabilization is used.
• The method demonstrates robustness and efficiency in simulating phase transitions in supercooled liquids and epitaxial growth, with minimal numerical diffusion.