Stochastic Wave Propagation in Maxwell’s Equations

Liu, Gi Ren

doi:10.1007/s10955-014-1148-y

Cited by 8 publications

(7 citation statements)

References 14 publications

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“…Taking the context of electromagnetism as an example, to model precise microscopic origins of randomness (the thermal motion of electrically charged microparticles), [12] established the theory of fluctuations of an electromagnetic field, which at the level of macroscopic view was via introducing fluctuation sources to obtain stochastic Maxwell equations. Based on this model, [14] proposed a method based on Wiener chaos expansion to determine the near field thermal radiation, and [10] described the fluctuation of the electromagnetic field using spectral representation. Without modeling the precise origins of randomness, rather assume that they lead to small stochastic variations of the coefficients of the equations, [7] studied the propagation of ultra-short solitons in a cubic nonlinear media, which is modeled by nonlinear Maxwell equations with stochastic variations of media; and assume that the externally imposed source is a random field, which is expressed by a Q-Wiener process, [4,9,11] dealt with the mathematical analysis of stochastic problems arising in the theory of electromagnetic in complex media, including well posedness, controllability and homogenisation.…”

Section: Introductionmentioning

confidence: 99%

Preservation of physical properties of stochastic Maxwell equations with additive noise via stochastic multi-symplectic methods

Chen

Hong

Zhang

2016

Journal of Computational Physics

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Stochastic Maxwell equations with additive noise are a system of stochastic Hamiltonian partial differential equations intrinsically, possessing the stochastic multi-symplectic conservation law. It is shown that the averaged energy increases linearly with respect to the evolution of time and the flow of stochastic Maxwell equations with additive noise preserves the divergence in the sense of expectation. Moreover, we propose three novel stochastic multi-symplectic methods to discretize stochastic Maxwell equations in order to investigate the preservation of these properties numerically. We make theoretical discussions and comparisons on all of the three methods to observe that all of them preserve the corresponding discrete version of the averaged divergence. Meanwhile, we obtain the corresponding dissipative property of the discrete averaged energy satisfied by each method. Especially, the evolution rates of the averaged energies for all of the three methods are derived which are in accordance with the continuous case. Numerical experiments are performed to verify our theoretical results.

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Section: Introductionmentioning

confidence: 99%

Preservation of physical properties of stochastic Maxwell equations with additive noise via stochastic multi-symplectic methods

Chen

Hong

Zhang

2016

Journal of Computational Physics

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“…Figure 2 presents the simulation of energy and the error of divergence by applying the proposed method (34) to two-dimensional stochastic Maxwell equations with additive noise (32). On one hand, from the left sub-figure, it is observed that the averaged energy (red line) is linear growth with respect to time and, on the other hand, it follows from the right sub-figure that the divergence error here is smaller than the case of 100 trajectories in [8], since the number of the sample path is increased.…”

Section: Theorem 42mentioning

confidence: 99%

“…In [21], the authors studied the approximate controllability of the stochastic Maxwell equations with an abstract approach and a constructive approach using a generalization of the Hilbert uniqueness method. Liu [32] utilized the spectral representation to describe the random electromagnetic fields and showed that the electromagnetic fields exhibit diffusive scaling limits in the sense of distributions when the stochastic source term is Gaussian.…”

Section: Introductionmentioning

confidence: 99%

A Review on Stochastic Multi-symplectic Methods for Stochastic Maxwell Equations

Zhang

Chen

Hong

et al. 2019

Commun. Appl. Math. Comput.

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Stochastic multi-symplectic methods are a class of numerical methods preserving the discrete stochastic multi-symplectic conservation law. These methods have the remarkable superiority to conventional numerical methods when applied to stochastic Hamiltonian partial differential equations (PDEs), such as long-time behavior, geometric structure preserving, and physical properties preserving. Stochastic Maxwell equations driven by either additive noise or multiplicative noise are a system of stochastic Hamiltonian PDEs intrinsically, which play an important role in fields such as stochastic electromagnetism and statistical radiophysics. Thereby, the construction and the analysis of various numerical methods for stochastic Maxwell equations which inherit the stochastic multi-symplecticity, the evolution laws of energy and divergence of the original system are an important and promising subject. The first stochastic multi-symplectic method is designed and analyzed to stochastic Maxwell equations by Hong et al. (A stochastic multi-symplectic scheme for stochastic Maxwell equations with additive noise. J. Comput. Phys. 268:255-268, 2014). Subsequently, there have been developed various stochastic multi-symplectic methods to solve stochastic Maxwell equations. In this paper, we make a review on these stochastic multi-symplectic methods for solving stochastic Maxwell equations driven by a stochastic process. Meanwhile, the theoretical results of well-posedness and conservation laws of the stochastic Maxwell equations are included.

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“…In [17], Liaskos et al studied the stochastic integrodifferential equations in Hilbert spaces, and they examined the well posedness for the Cauchy problem of the integrodifferential equations describing Maxwell equations. The random electromagnetic fields using the spectral representation is explored in [16], and the electromagnetic fields were coupled by Maxwell equations with a random source term. Finite element approximations of a class of nonlinear stochastic wave equations with multiplicative noise were recently investigated in [15].…”

Section: Introductionmentioning

confidence: 99%

Multi-symplectic discontinuous Galerkin methods for the stochastic Maxwell equations with additive noise

Sun,

Shu,

Xing

2021

Preprint

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One-and multi-dimensional stochastic Maxwell equations with additive noise are considered in this paper. It is known that such system can be written in the multi-symplectic structure, and the stochastic energy increases linearly in time. High order discontinuous Galerkin methods are designed for the stochastic Maxwell equations with additive noise, and we show that the proposed methods satisfy the discrete form of the stochastic energy linear growth property and preserve the multi-symplectic structure on the discrete level. Optimal error estimate of the semi-discrete DG method is also analyzed. The fully discrete methods are obtained by coupling with symplectic temporal discretizations. One-and two-dimensional numerical results are provided to demonstrate the performance of the proposed methods, and optimal error estimates and linear growth of the discrete energy can be observed for all cases.

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Stochastic Wave Propagation in Maxwell’s Equations

Cited by 8 publications

References 14 publications

Preservation of physical properties of stochastic Maxwell equations with additive noise via stochastic multi-symplectic methods

Preservation of physical properties of stochastic Maxwell equations with additive noise via stochastic multi-symplectic methods

A Review on Stochastic Multi-symplectic Methods for Stochastic Maxwell Equations

Multi-symplectic discontinuous Galerkin methods for the stochastic Maxwell equations with additive noise

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