A High‐Efficiency Linear Relaxation Scheme for Gradient Flows With Polynomial‐Type Nonlinear Potential Function
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| Format: | Artículo Open Access |
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2025
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| author | Xiaoqing Meng Aijie Cheng Zhengguang Liu |
| author_facet | Xiaoqing Meng Aijie Cheng Zhengguang Liu Xiaoqing Meng Aijie Cheng Zhengguang Liu |
| collection | Wiley Open Access |
| contents | A High‐Efficiency Linear Relaxation Scheme for Gradient Flows With Polynomial‐Type Nonlinear Potential Function Xiaoqing Meng Aijie Cheng Zhengguang Liu Numerical Methods for Partial Differential Equations ABSTRACTIn this paper, we establish a high‐efficiency new linear relaxation (NLR) scheme for gradient flows with polynomial‐type nonlinear potential functions. The Allen–Cahn model, Cahn–Hilliard model, Swift–Hohenberg model, and phase‐field crystal model are used as test platforms to verify the wide applicability and high efficiency of the proposed method. The NLR method consists of two key steps. Firstly, we discretize the equivalent transformed system on a staggered time grid by introducing polynomial‐type auxiliary variables. Secondly, we add an energy‐optimized technique to update the auxiliary variable to make the scheme possess unconditional energy stability. A major advantage over classical invariant energy quadratic and scalar auxiliary variable methods is that our method does not require a lower bound for a nonlinear function or its free energy. Rigorous theoretical analysis shows that the numerical schemes satisfy the unconditional energy stability at the discrete level in time, and maintain the mass conservation property of Cahn–Hilliard and phase‐field crystal models. In addition, ample numerical experiments demonstrate the ability of the NLR method in solving gradient flows. 10.1002/num.70039 http://onlinelibrary.wiley.com/termsAndConditions#vor |
| doi_str_mv | 10.1002/num.70039 |
| format | Artículo Open Access |
| id | wiley_oa_10_1002_num_70039 |
| institution | Wiley Open Access |
| license_str_mv | http://onlinelibrary.wiley.com/termsAndConditions#vor |
| publishDate | 2025 |
| publisher | Wiley |
| record_format | wiley_oa |
| spellingShingle | A High‐Efficiency Linear Relaxation Scheme for Gradient Flows With Polynomial‐Type Nonlinear Potential Function Xiaoqing Meng Aijie Cheng Zhengguang Liu Numerical Methods for Partial Differential Equations A High‐Efficiency Linear Relaxation Scheme for Gradient Flows With Polynomial‐Type Nonlinear Potential Function Xiaoqing Meng Aijie Cheng Zhengguang Liu Numerical Methods for Partial Differential Equations ABSTRACTIn this paper, we establish a high‐efficiency new linear relaxation (NLR) scheme for gradient flows with polynomial‐type nonlinear potential functions. The Allen–Cahn model, Cahn–Hilliard model, Swift–Hohenberg model, and phase‐field crystal model are used as test platforms to verify the wide applicability and high efficiency of the proposed method. The NLR method consists of two key steps. Firstly, we discretize the equivalent transformed system on a staggered time grid by introducing polynomial‐type auxiliary variables. Secondly, we add an energy‐optimized technique to update the auxiliary variable to make the scheme possess unconditional energy stability. A major advantage over classical invariant energy quadratic and scalar auxiliary variable methods is that our method does not require a lower bound for a nonlinear function or its free energy. Rigorous theoretical analysis shows that the numerical schemes satisfy the unconditional energy stability at the discrete level in time, and maintain the mass conservation property of Cahn–Hilliard and phase‐field crystal models. In addition, ample numerical experiments demonstrate the ability of the NLR method in solving gradient flows. 10.1002/num.70039 http://onlinelibrary.wiley.com/termsAndConditions#vor |
| title | A High‐Efficiency Linear Relaxation Scheme for Gradient Flows With Polynomial‐Type Nonlinear Potential Function |
| topic | Numerical Methods for Partial Differential Equations |
| url | https://onlinelibrary.wiley.com/doi/10.1002/num.70039 |