Research Article  Open Access
JunJie Cao, XiangGui Li, JingLiang Qiu, JingJing Zhang, "TimeCompact Scheme for the OneDimensional Dirac Equation", Discrete Dynamics in Nature and Society, vol. 2016, Article ID 3670139, 8 pages, 2016. https://doi.org/10.1155/2016/3670139
TimeCompact Scheme for the OneDimensional Dirac Equation
Abstract
Based on the Liealgebra, a new timecompact scheme is proposed to solve the onedimensional Dirac equation. This timecompact scheme is proved to satisfy the conservation of discrete charge and is unconditionally stable. The timecompact scheme is of fourthorder accuracy in time and spectral order accuracy in space. Numerical examples are given to test our results.
1. Introduction
The Dirac equation [1, 2], which is a spinor field equation and is applied to conquer the difficulty of the negative probability of KleinGordon equation, was proposed by the famous British physicist Dirac in 1928. In order to consider spin degree of freedom of electron, Dirac introduced multicomponent wave function and defined positive definite density of probability. The property of electron with high speed was discussed by using the Dirac equation. The fine structure of hydrogen atom was given when the Dirac equation was used to study the energy level distribution of hydrogen atom. In Dirac theory, spin1/2 and intrinsic magnetic moment of electron can be obtained. The Dirac equation predicts the existence of the antiparticle partner to the electron. Dirac equation was widely studied after producing of the graphene in the lab in 2003 [3, 4].
In this paper, we consider the onedimensional Dirac equation [5]whereis time, , and the magnetic potential and electronic potential are real. The complexvalued wave function of spinor field is . and are Pauli matrices is a dimensionless parameter, is the velocity of electron, and is the light speed. When , that is, the velocity of electron is far less than light velocity, there is , and (1) changes into the nonrelativistic model. On the contrary, it is relativistic model.
The Dirac equation (1) is dispersive and time symmetric. Here we introduce its position density and the total density which are defined aswhere is conjugate transpose of . Then, the charge of Dirac equation (1) is given as
For Dirac equation (1), it is hard to find the exact solution for general condition of electromagnetic potential. But for the special condition, such as coulomb problem, there was only nontrivial exact solution of Dirac equation [6]. The particle dynamics in relativistic quantum mechanics are described by using Dirac equation. Solution of Dirac equation is important in describing the nuclear shell structure [7]. In order to overcome the difficulty of solving analytically Dirac equation, many numerical methods, such as CrankNicolson finite difference method, timesplitting method [5], and pseudospectral method [8], have been used to solve numerically Dirac equation. To our best knowledge, the existing numerical methods for solving the Dirac equation have at most secondorder accuracy in time. In this paper, we give the timecompact scheme which uses the fewer time steps to reach the fourthorder accuracy to solve the onedimensional Dirac equation. This scheme may be extended to threedimensional Dirac equation.
The arrangement of the rest for this paper is organized as follows. The timecompact scheme with fourthorder accuracy is presented in Section 2. Numerical experiment is given to test the accuracy order and conservation of discrete charge in Section 3. Some conclusions are drawn in Section 4.
2. TimeCompact Scheme and Analysis
Recently, there has been growing interest in highorder compact method for solving partial differential equation, especially the timecompact methods [9–14], which can enhance the accuracy order in time. In addition, timecompact scheme is efficient for dealing with high frequency oscillation problem.
In this section, we will use the timecompact scheme with fourthorder accuracy to solve the Dirac equation (1) numerically; that is, we will discretize the equation by adopting timesplitting method in time and pseudospectral method in space. The aim is to enhance the accuracy order to the fourth order in time.
In practical computation, the computational domain is , which is large enough. We divide the interval into equal parts; then there are and ; here . Choose time step ; then ; here . That is,with the homogeneous boundary conditionand the initial condition
Setting (5) can be rewritten in form of
Obviously, (9) is a functional differential equation. So, we solve (1) in and obtain the formal solutionThe key of solving the solution of (10) is to give the approximate value of the operator .
For operators and , if they are commutative operators, the operator can be written as follows: In the general case, the operators and are noncommutative, and the equation above is not tenable. Assuming that and are noncommutative operators, according to the BakerCampbellHausdorff formula (BCH) [11], can be expressed in the form of a single exponential function aswhereHere we use the notation of the commutator
In order to solve the operator , based on Liealgebra and [12], one can obtain an approximate factorization in the form ofDenote; thenAccording to (12), we haveThenwithIn order to letmatch , we must choose and . Consider that the thirdorder term has the partial derivative of the unknown function (i.e.,), so we eliminate it by requiring . When , , , and , the other thirdorder term can be removed to yield the timecompact schemewhere
Sincewe haveTherefore,So we can obtainFrom time to time , for the operators , , and , we have three independent differential equations as follows:
According to the form of (26)–(28), (26) is discretized by the Fourier spectral method in space, (27) and (28) are functional differential system, and we can solve them analytically.
Next we begin to solve (10). In the first step, in , applying the pseudospectral method to solve (26) with initial value , we can obtain the solution via the inverse discrete Fourier transform as follows:where , , and the coefficient is obtained by using the discrete Fourier transformSubstituting (29) into (26) and combining with the initial condition , taking , we havewhereSubstituting into (29), we can obtain
In the second step, we solve (27) in by using variable separation method, the initial value is , and we get the solution where ,
The third step is to solve (26) in [] according to the first step with initial value . Thus, we can get the solution .
In the fourth step, we solve (28) by using the variable separation method in ] with the initial value solved from the third step and write the solution as where
The methods of the fifth, sixth, and seventh steps are obtained as the third, second, and first steps, respectively.
The timecompact scheme for solving the Dirac equation at can be obtained as follows:where , , andThe above scheme is explicit, and it is easy to iterate.
Theorem 1. The timecompact scheme (38a)–(38g) conserves the charge in the discretized level; that is,
Proof. Introduce the definition of the discrete inner product; that is,where and . Then we define the norm as follows:Equation (38a) is a Fourier transformation. From Parseval’s equality combining with (30), one obtains For (38b), we haveSimilarly, the proof of (38c), (38e), and (38g) and (38d) and (38f) is obtained as the proof of (38a) and (38b), respectively. That is,So, one can obtain the conservation of discrete charge
3. Numerical Example
In this section, we test the order of accuracy and stability of the timecompact scheme. In order to test the accuracy, we choose the electromagnetic potentials in (5) asand the initial condition as
We solve problem (5)–(7) numerically under the condition from to by using the timecompact scheme. The numerical results calculated by the timecompact scheme (TCS) with fourthorder accuracy and Strang splitting method (SSM) [5] with secondorder accuracy are listed in Table 1.

From Table 1, we can know that they have different accuracy order under the same condition, and the error of the timecompact scheme is much smaller than the error of Strang splitting method on the same row.
In every time step, the Strang splitting method with secondorder accuracy needs three steps, and the timecompact scheme with fourthorder accuracy needs seven steps; that is, the total number of steps of the secondorder accuracy scheme is of the fourthorder accuracy scheme. But, from the column of error and time, we can find that the fourthorder accuracy scheme has much bigger step size and costs less time than the secondorder accuracy when their errors have the same magnitude (such as ).
In order to test convergence of the algorithm with fourthorder , in the calculation, we verify the fourth order by taking enough small to examine the temporal fourth order. Figure 1 takes of enough small . We can know that the scheme in Section 2 is converged in maximum modulus [14], the convergence order for .
The discrete charge calculated by the timecompact scheme is given at different time. As the calculated results have shown, one can see that the timecompact scheme conserves the discrete charge.
From Theorem 1, we make the conclusion that the timecompact scheme is unconditionally stable.
When decreases, high frequency oscillation increases. The time evolution of for different is shown in Figure 2.
4. Conclusion
Based on Liealgebra, the timecompact scheme is presented for solving the onedimensional linear Dirac equation. Then we test whether the timecompact scheme has fourthorder accuracy in time and is proved to keep the conservation of discrete charge. From the numerical results, the timecompact scheme performs much better than the Strang splitting method in the error analysis, in terms of accuracy and efficiency. In addition, the timecompact scheme is unconditionally stable, and numerical experiment is presented to discuss the changes of the frequency oscillation with different . In the past years, much work has been done to investigate the dynamical properties of nonlinear Schrödinger equations and other physically important nonlinear wave equations [15–18]. Thus, in the future, we will develop the timecompact scheme to study the nonlinear Dirac equation [19, 20], which is a model of selfinteracting Dirac fermions in quantum field theory.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (no. 11671044) and in part by the Beijing Municipal Education Commission under Grant no. PXM2016_014224_000028.
References
 P. A. M. Dirac, “The quantum theory of the electron,” Proceedings of the Royal Society of London A: Mathematical and Physical Sciences, vol. 117, no. 778, pp. 610–624, 1928. View at: Publisher Site  Google Scholar
 P. A. M. Dirac, “A theory of electrons and protons,” Proceedings of the Royal Society of London A, vol. 126, no. 801, pp. 360–365, 1930. View at: Google Scholar
 K. S. Novoselov, A. K. Geim, S. V. Morozov et al., “Twodimensional gas of massless Dirac fermions in graphene,” Nature, vol. 438, no. 7065, pp. 197–200, 2005. View at: Publisher Site  Google Scholar
 D. A. Abanin, S. V. Morozov, L. A. Ponomarenko et al., “Giant nonlocality near the dirac point in graphene,” Science, vol. 332, no. 6027, pp. 328–330, 2011. View at: Publisher Site  Google Scholar
 W. Z. Bao, Y. Y. Cai, X. W. Jia, and Q. L. Tang, “Numerical methods and comparison for the Dirac equation in the nonrelativistic limit regime,” https://arxiv.org/abs/1504.02881 View at: Google Scholar
 H. Bahlouli, E. B. Choubabi, and A. Jellal, “Solution of onedimensional Dirac equation via poincare map,” Europhysics Letters, vol. 95, pp. 1305–1323, 2011. View at: Google Scholar
 M. Hamzavi and A. A. Rajabi, “Solution of Dirac equation with Killingbeck potential by using wave function ansatz method under spin symmetry limit,” Communications in Theoretical Physics, vol. 55, no. 1, pp. 35–37, 2011. View at: Publisher Site  Google Scholar  MathSciNet
 H. Wang, “A timesplitting spectral method for computing dynamics of spinor $F=1$ BoseEinstein condensates,” International Journal of Computer Mathematics, vol. 84, pp. 925–944, 2007. View at: Google Scholar
 J. L. Qiu, “Positive solutions for a nonlinear periodic boundaryvalue problem with a parameter,” Electronic Journal of Differential Equations, vol. 2012, no. 133, pp. 1–10, 2012. View at: Google Scholar
 R. D. Ruth, “A canonical integration technique,” IEEE Transactions on Nuclear Science, vol. 30, no. 4, pp. 2669–2671, 1983. View at: Publisher Site  Google Scholar
 M. Suzuki, “General theory of higherorder decomposition of exponential operators and symplectic integrators,” Physics Letters A, vol. 165, no. 56, pp. 387–395, 1992. View at: Publisher Site  Google Scholar  MathSciNet
 S. A. Chin, “Symplectic integrators from composite operator factorizations,” Physics Letters A, vol. 226, no. 6, pp. 344–348, 1997. View at: Publisher Site  Google Scholar  MathSciNet
 M. Suzuki, “Hybrid exponential product formulas for unbounded operators with possible applications to Monte Carlo simulations,” Physics Letters A, vol. 201, no. 56, pp. 425–428, 1995. View at: Publisher Site  Google Scholar  MathSciNet
 H. Yoshida, “Construction of higher order symplectic integrators,” Physics Letters A, vol. 150, no. 5–7, pp. 262–268, 1990. View at: Publisher Site  Google Scholar  MathSciNet
 D.S. Wang, X.H. Hu, J. Hu, and W. M. Liu, “Quantized quasitwodimensional BoseEinstein condensates with spatially modulated nonlinearity,” Physical Review A, vol. 81, no. 2, Article ID 025604, 4 pages, 2010. View at: Publisher Site  Google Scholar
 D.S. Wang, S. J. Yin, Y. Tian, and Y. Liu, “Integrability and bright soliton solutions to the coupled nonlinear Schrödinger equation with higherorder effects,” Applied Mathematics and Computation, vol. 229, pp. 296–309, 2014. View at: Publisher Site  Google Scholar  MathSciNet
 W. Z. Bao, Y. Y. Cai, and H. Q. Wang, “Efficient numerical methods for computing ground states and dynamics of dipolar BoseEinstein condensates,” Journal of Computational Physics, vol. 229, no. 20, pp. 7874–7892, 2010. View at: Publisher Site  Google Scholar  MathSciNet
 D.S. Wang and X. Wei, “Integrability and exact solutions of a twocomponent Kortewegde Vries system,” Applied Mathematics Letters, vol. 51, pp. 60–67, 2016. View at: Publisher Site  Google Scholar  MathSciNet
 F. W. Hehl and B. K. Datta, “Nonlinear spinor equation and asymmetric connection in general relativity,” Journal of Mathematical Physics, vol. 12, no. 7, pp. 1334–1339, 1971. View at: Publisher Site  Google Scholar  MathSciNet
 N. J. Popławski, “Nonsingular Dirac particles in spacetime with torsion,” Physics Letters B, vol. 690, no. 1, pp. 73–77, 2010. View at: Publisher Site  Google Scholar  MathSciNet
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Copyright © 2016 JunJie Cao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.