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Keywords = extended Fisher–Kolmogorov equation

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24 pages, 17859 KB  
Article
The Reduced-Dimension Method for Crank–Nicolson Mixed Finite Element Solution Coefficient Vectors of the Extended Fisher–Kolmogorov Equation
by Xiaohui Chang and Hong Li
Axioms 2024, 13(10), 710; https://doi.org/10.3390/axioms13100710 - 14 Oct 2024
Cited by 1 | Viewed by 881
Abstract
A reduced-dimension (RD) method based on the proper orthogonal decomposition (POD) technology and the linearized Crank–Nicolson mixed finite element (CNMFE) scheme for solving the 2D nonlinear extended Fisher–Kolmogorov (EFK) equation is proposed. The method reduces CPU runtime and error accumulation by reducing the [...] Read more.
A reduced-dimension (RD) method based on the proper orthogonal decomposition (POD) technology and the linearized Crank–Nicolson mixed finite element (CNMFE) scheme for solving the 2D nonlinear extended Fisher–Kolmogorov (EFK) equation is proposed. The method reduces CPU runtime and error accumulation by reducing the dimension of the unknown CNMFE solution coefficient vectors. For this purpose, the CNMFE scheme of the above EFK equation is established, and the uniqueness, stability and convergence of the CNMFE solutions are discussed. Subsequently, the matrix-based RDCNMFE scheme is derived by applying the POD method. Furthermore, the uniqueness, stability and error estimates of the linearized RDCNMFE solution are proved. Finally, numerical experiments are carried out to validate the theoretical findings. In addition, we contrast the RDCNMFE method with the CNMFE method, highlighting the advantages of the dimensionality reduction method. Full article
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8 pages, 763 KB  
Article
Explicit Integrating Factor Runge–Kutta Method for the Extended Fisher–Kolmogorov Equation
by Yanan Wang and Shuying Zhai
Math. Comput. Appl. 2023, 28(6), 110; https://doi.org/10.3390/mca28060110 - 22 Nov 2023
Cited by 2 | Viewed by 2178
Abstract
The extended Fisher–Kolmogorov (EFK) equation is an important model for phase transitions and bistable phenomena. This paper presents some fast explicit numerical schemes based on the integrating factor Runge–Kutta method and the Fourier spectral method to solve the EFK equation. The discrete global [...] Read more.
The extended Fisher–Kolmogorov (EFK) equation is an important model for phase transitions and bistable phenomena. This paper presents some fast explicit numerical schemes based on the integrating factor Runge–Kutta method and the Fourier spectral method to solve the EFK equation. The discrete global convergence of these new schemes is analyzed rigorously. Three numerical examples are presented to verify the theoretical analysis and the efficiency of the proposed schemes. Full article
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