A Second-Order Crank-Nicolson Leap-Frog Scheme for the Modified Phase Field Crystal Model with Long-Range Interaction

Area of Science:

• Computational Physics
• Materials Science
• Numerical Analysis

Background:

• The modified phase field crystal model (MPFC) with long-range interaction is crucial for simulating materials at the atomic scale.
• Efficient and stable numerical methods are required to solve the complex dynamics described by the MPFC model.

Purpose of the Study:

• To develop a fully discrete and decoupled numerical scheme for the MPFC model.
• To enhance computational efficiency and stability for simulations involving long-range interactions.

Main Methods:

• Construction of a Crank-Nicolson Leap-Frog (CNLF) scheme, treating stiff terms implicitly and non-stiff terms explicitly.
• Integration of the scalar auxiliary variable (SAV) method for explicit treatment of nonlinear potentials.
• Application of the Fourier spectral method for spatial discretization.

Main Results:

• The proposed CNLF scheme is fully discrete, second-order accurate, and decoupled.
• The scheme demonstrates unconditional stability, verified through theoretical analysis and numerical experiments.
• The method achieves high efficiency and accuracy in both 2D and 3D simulations.

Conclusions:

• The CNLF scheme offers a robust and efficient approach for solving the MPFC model with long-range interactions.
• The combination of CNLF and SAV methods results in a linear numerical scheme with constant coefficients, simplifying computations.
• The method's accuracy, efficiency, and stability are validated by extensive numerical experiments.