Discrete Dynamics in Nature and Society

Volume 2016 (2016), Article ID 3670139, 8 pages

http://dx.doi.org/10.1155/2016/3670139

## Time-Compact Scheme for the One-Dimensional Dirac Equation

School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China

Received 26 July 2016; Accepted 4 September 2016

Academic Editor: Cengiz Çinar

Copyright © 2016 Jun-Jie 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.

#### Abstract

Based on the Lie-algebra, a new time-compact scheme is proposed to solve the one-dimensional Dirac equation. This time-compact scheme is proved to satisfy the conservation of discrete charge and is unconditionally stable. The time-compact scheme is of fourth-order 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 Klein-Gordon 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, spin-1/2 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 one-dimensional Dirac equation [5]whereis time, , and the magnetic potential and electronic potential are real. The complex-valued 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 Crank-Nicolson finite difference method, time-splitting 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 second-order accuracy in time. In this paper, we give the time-compact scheme which uses the fewer time steps to reach the fourth-order accuracy to solve the one-dimensional Dirac equation. This scheme may be extended to three-dimensional Dirac equation.

The arrangement of the rest for this paper is organized as follows. The time-compact scheme with fourth-order 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. Time-Compact Scheme and Analysis

Recently, there has been growing interest in high-order compact method for solving partial differential equation, especially the time-compact methods [9–14], which can enhance the accuracy order in time. In addition, time-compact scheme is efficient for dealing with high frequency oscillation problem.

In this section, we will use the time-compact scheme with fourth-order accuracy to solve the Dirac equation (1) numerically; that is, we will discretize the equation by adopting time-splitting 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* Baker-Campbell-Hausdorff 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 Lie-algebra 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 third-order term has the partial derivative of the unknown function (i.e.,), so we eliminate it by requiring . When , , , and , the other third-order term can be removed to yield the time-compact 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 time-compact 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 time-compact 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 time-compact 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 time-compact scheme. The numerical results calculated by the time-compact scheme (TCS) with fourth-order accuracy and Strang splitting method (SSM) [5] with second-order accuracy are listed in Table 1.