We consider numerical approximations for a phase-field dendritic crystal growth model, which is a highly nonlinear system that couples the anisotropic Allen–Cahn type equation and the heat equation. By combining the stabilized-Invariant Energy Quadratization method with a novel decoupling technique, the scheme requires solving only a sequence of linear elliptic equations at each time step, making it the first, to the best of the author’s knowledge, totally decoupled, linear, unconditionally energy stable scheme for the model. We further prove the unconditional energy stability rigorously and present various numerical simulations to demonstrate the stability and accuracy.
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On a novel full decoupling, linear, second‐order accurate, and unconditionally energy stable numerical scheme for the anisotropic phase‐field dendritic crystal growth model
The anisotropic phase‐field dendritic crystal growth model is a highly nonlinear system that couples the anisotropic Allen–Cahn equation and the thermal equation together. Due to the high anisotropy and nonlinear couplings in the system, how to develop an accurate and efficient, especially a fully decoupled scheme, has always been a challenging problem. To solve the challenge, in this article, we construct a novel fully decoupled numerical scheme which is also linear, energy stable, and second‐order time accurate. The key idea to realize the full decoupling structure is to introduce an ordinary differential equation to deal with the nonlinear coupling terms satisfying the so‐called “zero‐energy‐contribution” property. This scheme is very effective and easy to implement since only a few fully decoupled elliptic equations with constant coefficients need to be solved at each time step. We rigorously prove the solvability of each step and the unconditional energy stability, and perform a large number of numerical simulations in 2D and 3D to demonstrate its stability and accuracy numerically.
more » « less- NSF-PAR ID:
- 10449325
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- International Journal for Numerical Methods in Engineering
- Volume:
- 122
- Issue:
- 16
- ISSN:
- 0029-5981
- Page Range / eLocation ID:
- p. 4129-4153
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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