Error analysis of stabilized semi-implicit method of Allen-Cahnequation

By Xiaofeng Yang. 2009

142 citations recorded by the source corpus.

We consider in this paper the stabilized semi-implicit (in time) scheme and the splitting scheme for the Allen-Cahn equation $\phi_t-\Delta\phi+$e$^-2f(\phi)=0$ arising from phase transitions in material science. For the stabilized first-order scheme, we show that it is unconditionally stable and the error bound depends on e-1 in some lower polynomial order using the spectrum estimate of [2, 10, 11]. In addition, the first- and second-order operator splitting schemes are proposed and the accuracy are tested and compared with the semi-implicit schemes numerically.

Subfields: Computational Theory and Mathematics, Materials Chemistry, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/62f09dc5de537a1bf74d41f868d967266632ec72

How researchers see this paper

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

In Computational Theory and Mathematics

1. Jiang Yang
2. Tao Tang
3. Jia Zhao
4. Qi Wang
5. Xiaofeng Yang
6. Jie Shen
7. Xinlong Feng
8. Junseok Kim

In Materials Chemistry

1. Xiaofeng Yang
2. Jiang Yang
3. Tao Tang
4. Jie Shen
5. Jia Zhao
6. Qi Wang
7. Xinlong Feng
8. Qiang Du

In Numerical Analysis

1. Jiang Yang
2. Tao Tang
3. Jie Shen
4. Qi Wang
5. Jia Zhao
6. Xiaofeng Yang
7. Xinlong Feng
8. Qiang Du

Citations in the corpus

• Implicit-Explicit Scheme for the Allen-Cahn Equation Preserves the Maximum Principle (2016, methodology): “For the Allen-Cahn equation, some recent stability analysis can be found in [4,6,12,14,15].”
• Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method (2017, methodology): “We note that a similar technique has been used in [3] to produce a linear scheme, where a zero term φ − φ is added, and the minuend is treated implicitly and the subtractor is treated explicitly, that is quite similar to the stabilization approach [31,38,39].”
• Numerical approximations of Allen-Cahn and Cahn-Hilliard equations (2010, background): “While it is possible to improve this bound to polynomial order in terms of ε−1 as is done in [12, 13, 15] and [24], this process is very technical so we decided not to carry it out as our main purpose in this paper is to study the stability properties of various schemes and to demonstrate that optim”
• Stabilized Crank-Nicolson/Adams-Bashforth Schemes for Phase Field Models (2013, methodology): “Yang gave the stabilized semi-implicit scheme and the splitting scheme for the Allen-Cahn equation [22].”
• Decoupled, Energy Stable Schemes for Phase-Field Models of Two-Phase Incompressible Flows (2015, background): “The difficulty (iii) is also well studied recently: two classes of methods, one based on stabilization [44, 40, 33, 42] and the other based on convex splitting [8, 9, 39, 6], have proved to be effective.”