A numerical method with particle conservation for the Maxwell-Dirac system

A numerical method is presented for solving the Maxwell-Dirac systems. The Maxwell equations with particle and current densities as the source terms are discretized explicitly. To guarantee the particle conservation, the Dirac equations coupled electromagnetic potentials are discretized by the time-splitting method and implicit finite difference. These numerical schemes are conservative in particle density and have second-order accuracy in time and space. One-dimensional numerical results are given to validate the accuracy and the conservation and three-dimensional examples are presented to describe dynamical behaviors of the Maxwell-Dirac system with several external potentials.

[1]  Conservation laws for the Maxwell-Dirac equations with dual Ohm’s law , 2007, math-ph/0703018.

[2]  V. Delgado.  Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac and other nonlinear Dirac equations in one space dimension , 1978 .

[3]  Maria Psarelli.  Maxwell–Dirac Equations in Four-Dimensional Minkowski Space , 2005 .

[4]  Nikolaos Bournaveas,et al.  Local existence for the Maxwell-Dirac equations in three space dimensions , 1996 .

[5]  Weizhu Bao,et al.  An efficient and stable numerical method for the Maxwell-Dirac system , 2004 .

[6]  Anadijiban Das,et al.  General solutions of Maxwell-Dirac equations in 1+1-dimensional space-time and a spatially confined solution , 1993 .

[7]  Semiclassical asymptotics for the Maxwell–Dirac system , 2003, math-ph/0305009.

[8]  Jie Shen,et al.  A Fourth-Order Time-Splitting Laguerre-Hermite Pseudospectral Method for Bose-Einstein Condensates , 2005, SIAM J. Sci. Comput..

[9]  A. Lisi.  A solitary wave solution of the Maxwell-Dirac equations , 1995 .

[10]  Bengt Fornberg,et al.  A split step approach for the 3-D Maxwell's equations , 2003 .

[11]  Localized solutions of the Dirac-Maxwell equations , 1995, hep-th/9510065.

[12]  A. Das.  An ongoing big bang model in the special relativistic Maxwell–Dirac equations , 1996 .

[13]  K. Nakanishi,et al.  Uniqueness of Finite Energy Solutions for Maxwell-Dirac and Maxwell-Klein-Gordon Equations , 2003 .

[14]  J. Chadam.  Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac equations in one space dimension , 1973 .

[15]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[16]  Chunxiong Zheng,et al.  A time-splitting spectral scheme for the Maxwell-Dirac system , 2005, 1205.0368.