Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method

By Xiaofeng Yang, Jia Zhao, Qi Wang. Journal of Computational Physics, 2017

340 citations recorded by the source corpus.

Abstract The Molecular Beam Epitaxial model is derived from the variation of a free energy, that consists of either a fourth order Ginzburg–Landau double well potential or a nonlinear logarithmic potential in terms of the gradient of a height function. One challenge in solving the MBE model numerically is how to develop proper temporal discretization for the nonlinear terms in order to preserve energy stability at the time-discrete level. In this paper, we resolve this issue by developing a first and second order time-stepping scheme based on the “Invariant Energy Quadratization” (IEQ) method. The novelty is that all nonlinear terms are treated semi-explicitly, and the resulted semi-discrete equations form a linear system at each time step. Moreover, the linear operator is symmetric positive definite and thus can be solved efficiently. We then prove that all proposed schemes are unconditionally energy stable. The semi-discrete schemes are further discretized in space using finite difference methods and implemented on GPUs for high-performance computing. Various 2D and 3D numerical examples are presented to demonstrate stability and accuracy of the proposed schemes.

Subfields: Aerospace Engineering, Computational Mechanics, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/c0c38bd36582bd02414f8361114d62a3e096b143

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Aerospace Engineering

1. Jie Shen
2. Jie Xu
3. Guangpu Zhu
4. Jiang Yang
5. Jun Yao
6. Jisheng Kou
7. Yushu Wu
8. Bowen Yao

In Computational Mechanics

1. Jie Xu
2. Jiang Yang
3. Jie Shen
4. Jisheng Kou
5. Guangpu Zhu
6. Jun Yao
7. Yushu Wu
8. Bowen Yao

In Materials Chemistry

1. Jie Xu
2. Jie Shen
3. Jiang Yang
4. Guangpu Zhu
5. Jun Yao
6. Jisheng Kou
7. Yushu Wu
8. Bowen Yao

Citations in the corpus

• The phase field method for geometric moving interfaces and their numerical approximations (2019, neutral): “Algorithm development and numerical analysis of three phase flows can be found in [414].”
• Thermodynamically consistent modelling of two-phase flows with moving contact line and soluble surfactants (2019, methodology): “Among these models, the Navier boundary condition (NBC) is a classical and natural boundary condition (Yang et al. 2017; Alpak, Samardžić & Frank 2018; Zampogna, Magnaudet & Bottaro 2019), and it has been used in simulations of contact line dynamics with insoluble surfactants.”
• The scalar auxiliary variable (SAV) approach for gradient flows (2018, background): “Yang and his collaborators [20,21,23,24,26,22, 19] made a big leap in generalizing the Lagrange multiplier approach to the so called invariant energy quadratization (IEQ) approach which is applicable to a large class of free energies.”