A Fourth Order Difference Scheme for the Maxwell Equations on Yee Grid

Fathy, Aly E.; Wang, Cheng; Wilson, J.; Yang, Songnan

doi:10.1142/s0219891608001623

Cited by 17 publications

(4 citation statements)

References 23 publications

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“…To minimize the discretization error, one can use a smaller pixel size, Δx, or higher-order approximations. Here, we use the fourth-order finite difference scheme 34 We discussed in Sec. 3 using MaxwellNet that was trained for cell phantoms to predict the scattered field for real cells.…”

Section: Discussionmentioning

confidence: 99%

“…To minimize the discretization error, one can use a smaller pixel size, Δx, or higher-order approximations. Here, we use the fourth-order finite difference scheme 34 in which convolutional kernels of [0,+1/24,−9/8,+9/8,−1/24] and [+1/24,−9/8,+9/8,−1/24, 0] are used for the calculation of the derivatives in Eq. (1).…”

Section: Appendix A: Calculation Of Physics-informed Lossmentioning

confidence: 99%

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Physics-informed neural networks for diffraction tomography

2022

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We propose a physics-informed neural network (PINN) as the forward model for tomographic reconstructions of biological samples. We demonstrate that by training this network with the Helmholtz equation as a physical loss, we can predict the scattered field accurately. It will be shown that a pretrained network can be fine-tuned for different samples and used for solving the scattering problem much faster than other numerical solutions. We evaluate our methodology with numerical and experimental results. Our PINNs can be generalized for any forward and inverse scattering problem.

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Section: Discussionmentioning

confidence: 99%

Section: Appendix A: Calculation Of Physics-informed Lossmentioning

confidence: 99%

Physics-informed neural networks for diffraction tomography

2022

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“…See the detailed derivations in the related references [24,25,35,47], etc. These long stencil fourth order finite difference approximations have been extensively applied to different types of partial differential equations (PDEs), such as incompressible Boussinesq equation [45,52], three-dimensional geophysical fluid models [44,48], the Maxwell equation [19].…”

Section: The Long Stencil Difference Operator and The Local Truncatio...mentioning

confidence: 99%

An energy stable fourth order finite difference scheme for the Cahn–Hilliard equation

Cheng

Feng

Wang

et al. 2019

Journal of Computational and Applied Mathematics

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147

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In this paper we propose and analyze an energy stable numerical scheme for the Cahn-Hilliard equation, with second order accuracy in time and the fourth order finite difference approximation in space. In particular, the truncation error for the long stencil fourth order finite difference approximation, over a uniform numerical grid with a periodic boundary condition, is analyzed, via the help of discrete Fourier analysis instead of the the standard Taylor expansion. This in turn results in a reduced regularity requirement for the test function. In the temporal approximation, we apply a second order BDF stencil, combined with a second order extrapolation formula applied to the concave diffusion term, as well as a second order artificial Douglas-Dupont regularization term, for the sake of energy stability. As a result, the unique solvability, energy stability are established for the proposed numerical scheme, and an optimal rate convergence analysis is derived in the ∞ (0, T ; 2 ) ∩ 2 (0, T ; H 2 h ) norm. A few numerical experiments are presented, which confirm the robustness and accuracy of the proposed scheme.Key words. Cahn-Hilliard equation, long stencil fourth order finite difference approximation, second order accuracy in time, energy stability, optimal rate convergence analysis, preconditioned steepest descent iteration

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“…This original JST scheme was proposed for the study of aerodynamical problems and computational fluid dynamics. Recently, its application has been extended to Maxwell's equations in non‐dispersive media [14].…”

Section: Introductionmentioning

confidence: 99%

Fourth‐order Jameson–Schmidt–Turkel FDTD scheme for non‐magnetised cold plasma

Prokopidis¹,

Zografopoulos

2020

Electron. lett.

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A fourth-order finite-difference time-domain (FDTD) scheme is proposed for the solution of Maxwell's equations in cold plasma (Drude medium), based on the multistage method of Jameson, Schmidt and Turkel, which was originally introduced in the framework of fluid dynamics. First, the system of governing differential equations is formed as a general first-order, operator-based approach, and then a four-stage algorithm is established. The accuracy of the method is verified in benchmark problems compared with analytical solutions and with the conventional second-order FDTD algorithm.

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A Fourth Order Difference Scheme for the Maxwell Equations on Yee Grid

Cited by 17 publications

References 23 publications

Physics-informed neural networks for diffraction tomography

Physics-informed neural networks for diffraction tomography

An energy stable fourth order finite difference scheme for the Cahn–Hilliard equation

Fourth‐order Jameson–Schmidt–Turkel FDTD scheme for non‐magnetised cold plasma

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