有界区域Weyl规范下具有周期间断系数Maxwell-Dirac系统多尺度算法
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摘要
Maxwell-Dirac系统及修正形式在拓扑绝缘体、石墨烯、超导等材料中有着十分广泛的应用,本文针对有界区域Weyl规范下具有周期间断系数Maxwell-Dirac系统,提出了该系统解的多尺度渐近展开式,结合时间分裂谱和自适应棱单元方法,发展了一类新型高效算法.数值计算结果表明该算法在处理上述时-空多尺度问题时十分有效.
The Maxwell-Dirac system has wide applications in materials science such as topological insulators, graphene, superconductors and so on. In this paper, we first present the homogenization method and the multiscale asymptotic method for the Maxwell-Dirac system with rapidly oscillating discontinuous coefficients in a bounded convex Lipschitz domain under the Weyl gauge. Based on the multiscale asymptotic expansions of the solution of the Maxwell-Dirac system, combining the time-splitting spectral method and the adaptive edge element method, we develop multiscale algorithms for solving the Maxwell-Dirac system with rapidly oscillating discontinuous coefficients and can deal with the multiscale problems with multiple time and space scales. Numerical examples are then carried out to validate the method presented in this paper.
The Maxwell-Dirac system has wide applications in materials science such as topological insulators, graphene, superconductors and so on. In this paper, we first present the homogenization method and the multiscale asymptotic method for the Maxwell-Dirac system with rapidly oscillating discontinuous coefficients in a bounded convex Lipschitz domain under the Weyl gauge. Based on the multiscale asymptotic expansions of the solution of the Maxwell-Dirac system, combining the time-splitting spectral method and the adaptive edge element method, we develop multiscale algorithms for solving the Maxwell-Dirac system with rapidly oscillating discontinuous coefficients and can deal with the multiscale problems with multiple time and space scales. Numerical examples are then carried out to validate the method presented in this paper.
引文
[1] Abenda S. Solitary waves for Maxwell-Dirac and Coulomb-Dirac models[C]. Annales de l'Institut Henri Poincare-A Physique Theorique. Paris:Gauthier-Villars, c1983-c1999., 1998, 68(2):229.
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[3] Bao W, Li X G. An efficient and stable numerical method for the Maxwell-Dirac system[J]. Journal of Computational Physics, 2004, 199(2):663-687.
[4] Bechouche P, Mauser N J.(Semi)-nonrelativistic limits of the Dirac equation with external timedependent electromagnetic field[J]. Communications in Mathematical Physics, 1998, 197(2):405-425. 1
[5] Bensoussan A, Lions J L, Papanicolaou G. Asymptotic analysis for periodic structures[M]. NorthHolland, Amsterdam, 1978.
[6] Bernevig B A, Hughes T L. Topological insulators and topological superconductors[M]. Princeton University Press, 2013.
[7] Bjorken J D, Drell S D. Relativistic Quantum Mechanics[M]. McGraw-Hill, 1965.
[8] Cao L Q, Zhang Y, Allegretto W and Lin Y P. Multiscale asymptotic method for Maxwell's equations in composite materials[J]. SIAM Journal on Numerical Analysis, 2010, 47(6):4257-4289.
[9] Chadam J M. Global solutions of the Cauchy problem for the(classical)coupled Maxwell-Dirac equations in one space dimension[J]. Journal of Functional Analysis, 1973, 13(2):173-184.
[10] Chadam J M, Glassey R T. On the Maxwell-Dirac equations with zero magnetic field and their solution in two space dimensions[J]. Journal of Mathematical Analysis and Applications, 1976,53(3):495-507.
[11] Cioranescu D, Donato P. An Introduction to Homogenization, volume 17 of Oxford Lecture Series in Mathematics and its Applications[J]. The Clarendon Press Oxford University Press, New York.2000, 10(31):106-109.
[12] D'Ancona P, Foschi D, Selberg S. Null structure and almost optimal local well-posedness of the Maxwell-Dirac system[J]. American Journal of Mathematics, 2010, 132(3):771-839.
[13]'Ancona P, Selberg S. Global well-posedness of the Maxwell-Dirac system in two space dimensions[J]. Journal of Functional Analysis, 2011, 260(8):2300-2365.
[14] Esteban M J, Georgiev V, SeréE. Stationary solutions of the Maxwell-Dirac and the KleinGordon-Dirac equations[J]. Calculus of Variations and Partial Differential Equations, 1996, 4(3):265-281.
[15] Esteban M J, Sere E. An overview on linear and nonlinear Dirac equations[J]. Discrete&Continuous Dynamical Systems-A, 2002, 8(2):381-397.
[16] Flato M, Taflin E, Simon J. On global solutions of the Maxwell-Dirac equations[J]. Communications in Mathematical Physics, 1987, 112(1):21-49.
[17] Georgiev V. Small amplitude solutions of the Maxwell-Dirac equations[J]. Indiana University Mathematics Journal, 1991:845-883.
[18] Glassey R T,Chadam J M. Properties of the solutions of the Cauchy problem for the(classical)coupled Maxwell-Dirac equations in one space dimension[J]. Proceedings of the American Mathematical Society, 1974, 43(2):373-378.
[19] Gross L. The cauchy problem for the coupled maxwell and dirac equations[J]. Communications on Pure&Applied Mathematics, 1966, 19(1):1-15.
[20] Huang Z, Jin S, Markowich P A, et al. A time-splitting spectral scheme for the Maxwell-Dirac system[J]. Journal of Computational Physics, 2005, 208(2):761-789.
[21] Katsnelson M I, Katsnel'son M I. Graphene:Carbon in Two Dimensions[M]. Cambridge University Press, 2012.
[22] Lisi A G. A solitary wave solution of the Maxwell-Dirac equations[J]. Journal of Physics A:Mathematical and General, 1995, 28(18):5385.
[23] McLachlan R I, Quispel G R W. Splitting methods[J]. Acta Numerica, 2002, 11(11):341-434.
[24] Oleinik O A, Shamaev A S, Yosifian G A. Mathematical Problems in Elasticity and Homogenization[M]. North-Holland, Amsterdam, 1992.
[25] Qi X L, Zhang S C. Topological insulators and superconductors[J]. Reviews of Modern Physics,2011, 83(4):1057.
[26] Raza H. Graphene Nanoelectronics:Metrology, Synthesis, Properties and Applications[M].Springer Science&Business Media, 2012.
[27] Roche S, Valenzuela S O. Topological Insulators:Fundamentals and Perspectives[M]. John Wiley&Sons, 2015.
[28] Shen S Q. Topological Insulators[M]. New York:Springer,2012.
[29] Shen S Q, Shan W Y, Lu H Z. Topological Insulator and the Dirac Equation[C]. Spin. World Scientific Publishing Company, 2011, 1(01):33-44.
[30] Sparber C, Markowich P. Semiclassical asymptotics for the Maxwell-Dirac system[J]. Journal of Mathematical Physics, 2003, 44(10):4555-4572.
[31] Strang G. On the construction and comparison of difference schemes[J]. SIAM Journal on Numerical Analysis, 1968, 5(3):506-517.
[32] Thaller B. The Dirac Equation[M]. Springer Science&Business Media, 2013.
[33] Wakano M. Intensely localized solutions of the classical Dirac-Maxwell field equations[J]. Progress of Theoretical Physics, 1966, 35(6):1117-1141.
[34] Zhang Y, Cao L Q, Wong Y S. Multiscale computations for 3d time-dependent maxwell's equations in composite materials[J]. SIAM Journal on Scientific Computing, 2010, 32(5):2560-2583.
[35] Zhang L, Cao L Q, and Luo J L. Multiscale analysis and computation for a stationary schr(o|¨)dingerpossion system in heterogeneous nanostructures[J]. Multiscale Modeling&Simulation, 2014, 12(1):
[2] Aharonov Y, Bohm D. Significance of electromagnetic potentials in the quantum theory[J]. Physical Review, 1959, 115(3):485-491.
[3] Bao W, Li X G. An efficient and stable numerical method for the Maxwell-Dirac system[J]. Journal of Computational Physics, 2004, 199(2):663-687.
[4] Bechouche P, Mauser N J.(Semi)-nonrelativistic limits of the Dirac equation with external timedependent electromagnetic field[J]. Communications in Mathematical Physics, 1998, 197(2):405-425. 1
[5] Bensoussan A, Lions J L, Papanicolaou G. Asymptotic analysis for periodic structures[M]. NorthHolland, Amsterdam, 1978.
[6] Bernevig B A, Hughes T L. Topological insulators and topological superconductors[M]. Princeton University Press, 2013.
[7] Bjorken J D, Drell S D. Relativistic Quantum Mechanics[M]. McGraw-Hill, 1965.
[8] Cao L Q, Zhang Y, Allegretto W and Lin Y P. Multiscale asymptotic method for Maxwell's equations in composite materials[J]. SIAM Journal on Numerical Analysis, 2010, 47(6):4257-4289.
[9] Chadam J M. Global solutions of the Cauchy problem for the(classical)coupled Maxwell-Dirac equations in one space dimension[J]. Journal of Functional Analysis, 1973, 13(2):173-184.
[10] Chadam J M, Glassey R T. On the Maxwell-Dirac equations with zero magnetic field and their solution in two space dimensions[J]. Journal of Mathematical Analysis and Applications, 1976,53(3):495-507.
[11] Cioranescu D, Donato P. An Introduction to Homogenization, volume 17 of Oxford Lecture Series in Mathematics and its Applications[J]. The Clarendon Press Oxford University Press, New York.2000, 10(31):106-109.
[12] D'Ancona P, Foschi D, Selberg S. Null structure and almost optimal local well-posedness of the Maxwell-Dirac system[J]. American Journal of Mathematics, 2010, 132(3):771-839.
[13]'Ancona P, Selberg S. Global well-posedness of the Maxwell-Dirac system in two space dimensions[J]. Journal of Functional Analysis, 2011, 260(8):2300-2365.
[14] Esteban M J, Georgiev V, SeréE. Stationary solutions of the Maxwell-Dirac and the KleinGordon-Dirac equations[J]. Calculus of Variations and Partial Differential Equations, 1996, 4(3):265-281.
[15] Esteban M J, Sere E. An overview on linear and nonlinear Dirac equations[J]. Discrete&Continuous Dynamical Systems-A, 2002, 8(2):381-397.
[16] Flato M, Taflin E, Simon J. On global solutions of the Maxwell-Dirac equations[J]. Communications in Mathematical Physics, 1987, 112(1):21-49.
[17] Georgiev V. Small amplitude solutions of the Maxwell-Dirac equations[J]. Indiana University Mathematics Journal, 1991:845-883.
[18] Glassey R T,Chadam J M. Properties of the solutions of the Cauchy problem for the(classical)coupled Maxwell-Dirac equations in one space dimension[J]. Proceedings of the American Mathematical Society, 1974, 43(2):373-378.
[19] Gross L. The cauchy problem for the coupled maxwell and dirac equations[J]. Communications on Pure&Applied Mathematics, 1966, 19(1):1-15.
[20] Huang Z, Jin S, Markowich P A, et al. A time-splitting spectral scheme for the Maxwell-Dirac system[J]. Journal of Computational Physics, 2005, 208(2):761-789.
[21] Katsnelson M I, Katsnel'son M I. Graphene:Carbon in Two Dimensions[M]. Cambridge University Press, 2012.
[22] Lisi A G. A solitary wave solution of the Maxwell-Dirac equations[J]. Journal of Physics A:Mathematical and General, 1995, 28(18):5385.
[23] McLachlan R I, Quispel G R W. Splitting methods[J]. Acta Numerica, 2002, 11(11):341-434.
[24] Oleinik O A, Shamaev A S, Yosifian G A. Mathematical Problems in Elasticity and Homogenization[M]. North-Holland, Amsterdam, 1992.
[25] Qi X L, Zhang S C. Topological insulators and superconductors[J]. Reviews of Modern Physics,2011, 83(4):1057.
[26] Raza H. Graphene Nanoelectronics:Metrology, Synthesis, Properties and Applications[M].Springer Science&Business Media, 2012.
[27] Roche S, Valenzuela S O. Topological Insulators:Fundamentals and Perspectives[M]. John Wiley&Sons, 2015.
[28] Shen S Q. Topological Insulators[M]. New York:Springer,2012.
[29] Shen S Q, Shan W Y, Lu H Z. Topological Insulator and the Dirac Equation[C]. Spin. World Scientific Publishing Company, 2011, 1(01):33-44.
[30] Sparber C, Markowich P. Semiclassical asymptotics for the Maxwell-Dirac system[J]. Journal of Mathematical Physics, 2003, 44(10):4555-4572.
[31] Strang G. On the construction and comparison of difference schemes[J]. SIAM Journal on Numerical Analysis, 1968, 5(3):506-517.
[32] Thaller B. The Dirac Equation[M]. Springer Science&Business Media, 2013.
[33] Wakano M. Intensely localized solutions of the classical Dirac-Maxwell field equations[J]. Progress of Theoretical Physics, 1966, 35(6):1117-1141.
[34] Zhang Y, Cao L Q, Wong Y S. Multiscale computations for 3d time-dependent maxwell's equations in composite materials[J]. SIAM Journal on Scientific Computing, 2010, 32(5):2560-2583.
[35] Zhang L, Cao L Q, and Luo J L. Multiscale analysis and computation for a stationary schr(o|¨)dingerpossion system in heterogeneous nanostructures[J]. Multiscale Modeling&Simulation, 2014, 12(1):