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Keywords = Crank–Nicolson Leap-Frog

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26 pages, 657 KB  
Article
Spectral Galerkin Methods for Riesz Space-Fractional Convection–Diffusion Equations
by Xinxia Zhang, Jihan Wang, Zhongshu Wu, Zheyi Tang and Xiaoyan Zeng
Fractal Fract. 2024, 8(7), 431; https://doi.org/10.3390/fractalfract8070431 - 22 Jul 2024
Cited by 5 | Viewed by 1686
Abstract
This paper applies the spectral Galerkin method to numerically solve Riesz space-fractional convection–diffusion equations. Firstly, spectral Galerkin algorithms were developed for one-dimensional Riesz space-fractional convection–diffusion equations. The equations were solved by discretizing in space using the Galerkin–Legendre spectral approaches and in time using [...] Read more.
This paper applies the spectral Galerkin method to numerically solve Riesz space-fractional convection–diffusion equations. Firstly, spectral Galerkin algorithms were developed for one-dimensional Riesz space-fractional convection–diffusion equations. The equations were solved by discretizing in space using the Galerkin–Legendre spectral approaches and in time using the Crank–Nicolson Leap-Frog (CNLF) scheme. In addition, the stability and convergence of semi-discrete and fully discrete schemes were analyzed. Secondly, we established a fully discrete form for the two-dimensional case with an additional complementary term on the left and then obtained the stability and convergence results for it. Finally, numerical simulations were performed, and the results demonstrate the effectiveness of our numerical methods. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)
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15 pages, 552 KB  
Article
An Explicit–Implicit Spectral Element Scheme for the Nonlinear Space Fractional Schrödinger Equation
by Zeting Liu, Baoli Yin and Yang Liu
Fractal Fract. 2023, 7(9), 654; https://doi.org/10.3390/fractalfract7090654 - 30 Aug 2023
Viewed by 1398
Abstract
In this paper, we solve the space fractional nonlinear Schrödinger equation (SFNSE) by developing an explicit–implicit spectral element scheme, which is formulated based on the Legendre spectral element approximation in space and the Crank–Nicolson leap frog (CNLF) difference discretization in time. Both mass [...] Read more.
In this paper, we solve the space fractional nonlinear Schrödinger equation (SFNSE) by developing an explicit–implicit spectral element scheme, which is formulated based on the Legendre spectral element approximation in space and the Crank–Nicolson leap frog (CNLF) difference discretization in time. Both mass and energy conservative properties are discussed for the spectral element scheme. Numerical stability and convergence of the scheme are proved. Numerical experiments are performed to confirm the high accuracy and efficiency of the proposed numerical scheme. Full article
(This article belongs to the Section Numerical and Computational Methods)
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18 pages, 4385 KB  
Article
A Second-Order Crank–Nicolson Leap-Frog Scheme for the Modified Phase Field Crystal Model with Long-Range Interaction
by Chunya Wu, Xinlong Feng and Lingzhi Qian
Entropy 2022, 24(11), 1512; https://doi.org/10.3390/e24111512 - 23 Oct 2022
Cited by 2 | Viewed by 2147
Abstract
In this paper, we construct a fully discrete and decoupled Crank–Nicolson Leap-Frog (CNLF) scheme for solving the modified phase field crystal model (MPFC) with long-range interaction. The idea of CNLF is to treat stiff terms implicity with Crank–Nicolson and to treat non-stiff terms [...] Read more.
In this paper, we construct a fully discrete and decoupled Crank–Nicolson Leap-Frog (CNLF) scheme for solving the modified phase field crystal model (MPFC) with long-range interaction. The idea of CNLF is to treat stiff terms implicity with Crank–Nicolson and to treat non-stiff terms explicitly with Leap-Frog. In addition, the scalar auxiliary variable (SAV) method is used to allow explicit treatment of the nonlinear potential, then, these technique combines with CNLF can lead to the highly efficient, fully decoupled and linear numerical scheme with constant coefficients at each time step. Furthermore, the Fourier spectral method is used for the spatial discretization. Finally, we show that the CNLF scheme is fully discrete, second-order decoupled and unconditionally stable. Ample numerical experiments in 2D and 3D are provided to demonstrate the accuracy, efficiency, and stability of the proposed method. Full article
(This article belongs to the Special Issue Finite Element Methods for the Navier-Stokes Equations and MHD Equations)
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19 pages, 1589 KB  
Article
Analysis of Metal Oxide Varistor Arresters for Protection of Multiconductor Transmission Lines Using Unconditionally-Stable Crank–Nicolson FDTD
by Erika Stracqualursi, Rodolfo Araneo, Giampiero Lovat, Amedeo Andreotti, Paolo Burghignoli, Jose Brandão Faria and Salvatore Celozzi
Energies 2020, 13(8), 2112; https://doi.org/10.3390/en13082112 - 24 Apr 2020
Cited by 11 | Viewed by 4040
Abstract
Surge arresters may represent an efficient choice for limiting lightning surge effects, significantly reducing the outage rate of power lines. The present work firstly presents an efficient numerical approach suitable for insulation coordination studies based on an implicit Crank–Nicolson finite difference time domain [...] Read more.
Surge arresters may represent an efficient choice for limiting lightning surge effects, significantly reducing the outage rate of power lines. The present work firstly presents an efficient numerical approach suitable for insulation coordination studies based on an implicit Crank–Nicolson finite difference time domain method; then, the IEEE recommended surge arrester model is reviewed and implemented by means of a local implicit scheme, based on a set of non-linear equations, that are recast in a suitable form for efficient solution. The model is proven to ensure robustness and second-order accuracy. The implementation of the arrester model in the implicit Crank–Nicolson scheme represents the added value brought by the present study. Indeed, its preserved stability for larger time steps allows reducing running time by more than 60 % compared to the well-known finite difference time domain method based on the explicit leap-frog scheme. The reduced computation time allows faster repeated solutions, which need to be looked for on assessing the lightning performance (randomly changing, parameters such as peak current, rise time, tail time, location of the vertical leader channel, phase conductor voltages, footing resistance, insulator strength, etc. would need to be changed thousands of times). Full article
(This article belongs to the Special Issue Power and Signal Transmission Lines Modeling and Large Network Analysis)
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