Communications on Applied Mathematics and Computation >

2019 , Vol. 1 >Issue 3: 467 - 501

DOI: https://doi.org/10.1007/s42967-019-00017-w

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

Expand
  • 1 School of Mathematical Science, China University of Mining and Technology, Beijing 100083, China;
    2 LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
    3 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China;
    4 Institute of Applied Physics and Computational Mathematics, Beijing 100094, China

Received date: 2018-06-27

  Revised date: 2018-12-16

  Online published: 2019-09-09

Supported by

The research of L. Zhang was supported by the NNSFC (NOs. 11601514, 11771444, and 11801556), the research of C. Chen and J. Hong were supported by the NNSFC (NOs. 91630312, 11711530071, and 11871068), the research of L. Ji was supported by the NNSFC (NOs. 11601032, and 11471310).

Fold

Abstract

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.

Key words： Stochastic multi-symplectic methods; Stochastic Hamiltonian partial differential equations; Stochastic Maxwell equations; Structure-preserving methods

Cite this article

Liying Zhang, Chuchu Chen, Jialin Hong, Lihai Ji . A Review on Stochastic Multi-symplectic Methods for Stochastic Maxwell Equations[J]. Communications on Applied Mathematics and Computation, 2019 , 1(3) : 467 -501 . DOI: 10.1007/s42967-019-00017-w

References

1. Andrews, L.C., Phillips, R.L.:Laser Beam Propagation Through Random Media. SPIE Press, Bellingham (2005)
2. Andreas, S.:Locally implicit time integration for linear Maxwell's equations, Ph. D. Thesis, Karlsruhe (2017)
3. Bridges, T., Reich, S.:Multi-symplectic integrators:numerical schemes for Hamiltonian PDEs that conserve symplecticity. Phys. Lett. A 284, 184-193 (2001)
4. Bridges, T., Reich, S.:Numerical methods for Hamiltonian PDEs. J. Phys. A Math. Gen. 39, 5287- 5320 (2006)
5. Badieirostami, M., Adibi, A., Zhou, H., Chow, S.:Wiener chaos expansion and simulation of electromagnetic wave propagation excited by a spatially incoherent source. Multiscale Model. Simul. 8, 591-604 (2010)
6. Benner, P., Schneider, J.:Uncertainty quantification for Maxwell's equations using stochastic collocation and model order reduction. Int. J. Uncertain. Quantif. 5, 195-208 (2015)
7. Chan, D.L.C., Soljacic, M., Joannopoulos, J.D.:Direct calculation of thermal emission for three dimensionally periodic photonic crystal slabs. Phys. Rev. E 74, 036615 (2006)
8. Chen, C., Hong, J., Zhang, L.:Preservation of physical properties of stochastic Maxwell equations with additive noise via stochastic multi-symplectic methods. J. Comput. Phys. 306, 500-519 (2016)
9. Chen, C., Hong, J.:Symplectic Runge-Kutta semidiscretization for stochastic Schrödinger equation. SIAM. J. Numer. Anal. 54, 2569-2593 (2016)
10. Chen, C., Hong, J., Ji, L.:Mean-square convergence of a semidiscrete scheme for stochastic Maxwell equations. SIAM. J. Numer. Anal. 57, 728-750 (2019)
11. Chen, C., Hong, J., Ji, L.:Runge-Kutta semidiscretizations for stochastic Maxwell equations with additive noise. SIAM J. Numer. Anal. 57, 702-727 (2019)
12. Deng, J., Anton, C., Wong, Y.:High-order symplectic schemes for stochastic Hamiltonian systems. Commun. Comput. Phys. 16, 169-200 (2014)
13. Eyges, L.:The Classical Electromagnetic Fields. Addison-Wesley, New York (1972)
14. Engel, K.J., Nagel, R.:One-Parameter Semigroups for Linear Evolution Equations. Springer, Berlin (2000)
15. Francoeur, M., Mengüç, M.:Role of fluctuational electrodynamics in near-field radiative heat transfer. J. Quant. Spectrosc. Radiat. Transf. 109, 280-293 (2008)
16. Feng, K., Qin, M.:Symplectic Geometric Algorithms for Hamiltonian Systems. Springer-Verlag/Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Berlin/Hangzhou (2010)
17. Fouque, J.P., Garnier, J., Papanicolaou, G., Sølna, K.:Wave Propagation and Time Reversal in Randomly Layered Media. Springer, New York (2007)
18. Field, T.R.:Electromagnetic Scattering from Random Media. Oxford University Press, Oxford (2009)
19. Goldsmith, A.:Wireless Communications. Cambridge University Press, Cambridge (2005)
20. Hairer, E., Lubich, C., Wanner, G.:Geometric Numerical Integration:Structure Preserving Algorithms for Ordinary Differential Equations. Springer, Berlin (2002)
21. Horsin, T., Stratis, I.G., Yannacopoulos, A.N.:On the approximate controllability of the stochastic Maxwell equations. IMA J. Math. Control. I. 27, 103-118 (2010)
22. Hong, J., Ji, L., Zhang, L.:A stochastic multi-symplectic scheme for stochastic Maxwell equations with additive noise. J. Comput. Phys. 268, 255-268 (2014)
23. Hong, J., Ji, L., Zhang, L., Cai, J.:An energy-conserving method for stochastic Maxwell equations with multiplicative noise. J. Comput. Phys. 351, 216-229 (2017)
24. Hornung, L.:Strong solutions to a nonlinear stochastic Maxwell equation with a retarded material law. J. Evol. Equ. 18, 1427-1469 (2018)
25. Ikeda, N., Watanabe, S.:Stochastic Differential Equations and Diffusion Processes. North-Holland Publishing Company, Amsterdam (1981)
26. Jackson, J.D.:Classical Electrodynamics, 3rd edn. Wiley, New York (1999)
27. Jiang, S., Wang, L., Hong, J.:Stochastic multi-symplectic integrator for stochastic nonlinear Schrödinger equation. Commun. Comput. Phys. 14, 393-411 (2013)
28. Jung, C.:Evolution of probability distribution in time for solutions of hyperbolic equations. J. Sci. Comput. 41, 13-48 (2009)
29. Jung, C., Kwon, B., Mahalov, A., Nguyen, T.:Maxwell solutions in media with multiple random interfaces. Int. J. Numer. Anal. Mod. 11, 194-213 (2014)
30. Karlsson, A., Kristensson, G.:Constitutive relations, dissipation and reciprocity for the Maxwell equations in the time domain. J. Electrom. Waves Appl. 6, 537-551 (1992)
31. Liaskos, K.B., Stratis, I.G., Yannacopoulos, A.N.:Stochastic integrodifferential equations in Hilbert spaces with applications in electromagnetics. J. Integral Equations Appl. 22, 559-590 (2010)
32. Liu, G.:Stochastic wave propagation in Maxwell's equations. J. Stat. Phys. 158, 1126-1146 (2015)
33. Li, J., Fang, Z., Lin, G.:Regularity analysis of metamaterial Maxwell equations with random coefficients and initial conditions. Comput. Methods Appl. Mech. Engrg. 335, 24-51 (2018)
34. Marsden, J., Patrick, G., Shkoller, S.:Multi-symplectic geometry, variational integrators, and nonlinear PDEs. Comm. Math. Phys. 199, 351-395 (1998)
35. Milstein, G.N., Repin, Y.M., Tretyakov, M.V.:Numerical methods for stochastic systems preserving symplectic structure. SIAM J. Numer. Anal. 40, 1583-1604 (2002)
36. Milstein, G.N., Repin, Y.M., Tretyakov, M.V.:Symplectic integration of Hamiltonian systems with additive noise. SIAM J. Numer. Anal. 39, 2066-2088 (2002)
37. Milstein, G.N., Tretyakov, M.V.:Stochastic Numerics for Mathematical Physics. Springer, Berlin (2004)
38. Milton, G.W.:The Theory of Composites. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2004)
39. Moscoso, M.:Polarization-Based Optical Imaging. In:Bonilla, L.L. (eds.) Inverse Problems and Imaging, Lecture Notes in Mathematics, vol. 1943. Springer, Berlin (2008)
40. Da Prato, G., Zabczyk, J.:Stochastic Equations in Infinite Dimensions, 2nd edn. Cambridge University Press, Cambridge (2014)
41. Rytov, S.M., Kravtsov, Y.A., Tatarskii, V.I.:Principles of Statistical Radiophysics 3:Elements of Random Fields. Springer, Berlin (1989)
42. Roach, G.F., Stratis, I.G., Yannacopoulos, A.N.:Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics. Princeton Series in Applied Mathematics, Princeton University Press, Princeton (2012)
43. Skolnik, M.I.:Radar Handbook, 3rd edn. McGraw-Hill, New York (2008)
44. Süß, A., Waurick, M.:A solution theory for a general class of SPDEs:Stoch PD. Anal. Comp. 5, 278-318 (2017)
45. Wang, L.:Variational integrators and generating functions for stochastic Hamiltonian systems, Ph. D. Thesis, Universitätsverlag Karlsruhe, (2007)
46. Wang, L., Hong, J.:Generating functions for stochastic symplectic methods. Discrete Contin. Dyn. Syst. 34, 1211-1228 (2014)
47. Wang, Y., Hong, J.:Multi-symplectic algorithms for Hamiltonian partial differential equations. Commun. Appl. Math. Comput. 27, 163-230 (2013)
48. Zhang, K.:Numerical studies of some stochastic partial differential equations. Ph. D. Thesis, The Chinese University of Hong Kong, China, (2008)
49. Zhou, W., Zhang, J., Hong, J., Song, S.:Stochastic symplectic Runge-Kutta methods for the strong approximation of Hamiltonian systems with additive noise. J. Comput. Appl. Math. 325, 134-148 (2017)
Options
Outlines

/

〈 〉

Copyright © Editorial By Communications on Applied Mathematics and Computation
沪ICP备09014157