A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation

Transcription

1 Research in Mathematical Sciences manuscript No. will be inserted by the editor A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation Weizhu Bao Jia Yin Received: date / Accepted: date Abstract We propose a new fourth-order compact time-splitting S 4c Fourier pseudospectral method for the Dirac equation by splitting the Dirac equation into two parts together with using the double commutator between them to integrate the Dirac equation at each time interval. The method is explicit, fourth-order in time and spectral order in space. It is unconditional stable and conserves the total probability in the discretized level. It is called a compact time-splitting method since, at each time step, the number of sub-steps in S 4c is much less than those of the standard fourth-order splitting method and the fourth-order partitioned Runge-Kutta splitting method. Another advantage of S 4c is that it avoids to use negative time steps in integrating sub-problems at each time interval. Comparison between S 4c and many other existing time-splitting methods for the Dirac equation are carried out in terms of accuracy and efficiency as well as long time behavior. Numerical results demonstrate the advantage in terms of efficiency and accuracy of the proposed S 4c. Finally we report the spatial/temporal resolutions of S 4c for the Dirac equation in different parameter regimes including the nonrelativistic limit regime, the semiclassical limit regime, and the simultaneously nonrelativisic and massless limit regime. Keywords Dirac equation fourth-order compact time-splitting double commutator probability conservation nonrelativistic limit regime semiclassical limit regime This work was partially supported by the Ministry of Education of Singapore grant R MOE2015-T Weizhu Bao Department of Mathematics, National University of Singapore, Singapore , Singapore matbaowz@nus.edu.sg URL: bao/ Jia Yin Corresponding author NUS Graduate School for Integrative Sciences and Engineering NGS, National University of Singapore, Singapore , Singapore e @u.nus.edu

2 2 Weizhu Bao, Jia Yin 1 Introduction The Dirac equation was proposed by British physicist Paul Dirac in 1928 in order to integrate special relativity with quantum mechanics [29]. It successfully solved the problem that the probability density could be negative in the Klein-Gordon equation proposed by Oskar Klein and Walter Gordon in 1926 [28]. The Dirac equation describes the motion of relativistic spin- 1/2 massive particles, such as electrons and quarks. It fully explained the hydrogen spectrum and predicted the existence of antimatter. Recently, the Dirac equation has been extensively adopted to investigate theoretically the structures and/or dynamical properties of graphene and graphite as well as other two-dimensional 2D materials [1, 33, 49, 48], and to study the relativistic effects in molecules in super intense lasers, e.g., attosecond lasers [18,36]. Consider the Dirac equation with electromagnetic potentials in three spatial dimensions 3D [29 31, 60] i h t Ψ = ic h 3 α j j + mc 2 β Ψ + e VxI 4 3 A j xα j Ψ, x R 3, 1.1 where t is time, x=x 1,x 2,x 3 T or x=x,y,z T is the spatial coordinate, Ψ := Ψt,x= ψ 1 t,x, ψ 2 t,x,ψ 3 t,x,ψ 4 t,x T C 4 is the complex-valued spinor wave function, and j represents x j for j = 1,2,3. The constants used in the equation are: i = 1, h is the Planck constant, m is the mass, c is the speed of light and e is the unit charge. In addition, V :=Vx is the time-independent electric potential and A := Ax=A 1 x,a 2 x,a 3 x T stands for the time-independent magnetic potential, which are all real-valued given functions. Finally, the 4 4 matrices β and α j j= 1,2,3 are the Dirac representation matrices of the four-dimensional Clifford algebra, which are given as I σ j β =, α j =,,2,3, I 2 σ j 0 where I n is the n n identity matrix and σ j,2,3 are the Pauli matrices defined as: i σ 1 =, σ 2 =, σ 3 = 1 0 i In order to nondimensionalize the Dirac equation 1.1, we take x= x x s, t = t t s, Ṽ = V A s, Ã= A A s, Ψ t, x= Ψt,x ψ s, 1.4 where x s, t s and m s are length unit, time unit and mass unit, respectively, to be taken for the nondimensionalization of the Dirac equation 1.1. Plugging 1.4 into 1.1 and taking ψ s = xs 3/2 and A s = m sx 2 s, after some simplification and then removing all, we obtain the ets 2

3 A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation 3 dimensionless Dirac equation in 3D iδ t Ψ = i δ 3 α j j + ν 2 β Ψ + VxI 4 where the three dimensionless parameters 0 <, δ, ν 1 are given as 3 A j xα j Ψ, x R 3, 1.5 = x s t s c = v s c, δ = ht s m s x 2, ν = m, 1.6 s m s with v s = x s /t s the velocity unit for nondimensionalization. In fact, here represents the ratio between the wave velocity and the speed of light, i.e. it is inversely proportional to the speed of light, δ stands for the scaled Planck constant and ν is the ratio between the mass of the particle and the mass unit taken for the nondimensionalization. As discussed in [9], under proper assumption on the electromagnetic potentials Vx and Ax, the Dirac equation 1.5 in 3D could be reduced to two dimensions 2D and one dimension 1D. Specifically, the Dirac equation in 2D has been widely applied to model the electron structure and dynamical properties of graphene and other 2D materials as they share the same dispersion relation on certain points called Dirac points [33 35, 48]. In fact, the Dirac equation 1.5 in 3D and its dimension reduction in 2D and 1D can be formulated in a unified way in d-dimensions d = 1, 2, 3 as iδ t Ψ = i δ d α j j + ν 2 β Ψ+ VxI 4 d A j xα j Ψ, x R d, 1.7 where x=x 1,x 2 T or x=x,y T in 2D and x=x 1 or x=x in 1D. To study the dynamics of the Dirac equation 1.7, the initial condition is usually taken as Ψt = 0,x= Ψ 0 x, x R d. 1.8 The Dirac equation 1.7 with 1.8 is dispersive, time-symmetric, and it conserves the total probability [9] Ψt, 2 := dx= R d Ψt,x 2 R d and the energy [9] EΨt, := R d i δ d 4 ψ j t,x 2 dx Ψ0, 2 = Ψ 0 2, t 0, 1.9 Ψ α j j Ψ + ν 2Ψ βψ+vx Ψ 2 d A j xψ α j Ψ EΨ 0, t 0, 1.10 where Ψ = Ψ T with f denoting the complex conjugate of f. dx

4 4 Weizhu Bao, Jia Yin Introduce the total probability densityρ := ρt, x as ρt,x= 4 ρ j t,x= Ψt,x Ψt,x, x R d, 1.11 where the probability density ρ j := ρ j t,x of the j-th j = 1,2,3,4 component is defined as and the current density Jt,x=J 1 t,x,...,j d t,x T as ρ j t,x= ψ j t,x 2, x R d, 1.12 J l t,x= 1 Ψt,x α l Ψt,x, l = 1,...,d, 1.13 then the following conservation law can be obtained from the Dirac equation 1.7 [9] t ρt,x+ Jt,x=0, x R d, t If the electric potential V is perturbed by a real constant V 0, i.e., V V +V 0, then the solution Ψt,x e i V 0 t δ Ψt,x, which implies that the probability density of each component ρ j,2,3,4 and the total probability density ρ are all unchanged. In addition, when d= 1, if the magnetic potential A 1 is perturbed by a real constant A 0 1, i.e., A 1 A 1 +A 0 1, then the solution Ψt,x e i A0 1 t δ α 1 Ψt,x, which implies that only the total probability density ρ is unchanged; however, this property is unfortunately not valid in 2D and 3D. Furthermore, if the external electromagnetic potentials are all real constants, i.e. Vx V 0 and A j x A 0 j j= 1,...,d with A0 =A 0 1,...,A0 d T, the Dirac equation 1.7 admits the plane wave solution Ψt,x=Be ik x ω δ t with ω the time frequency, B R 4 the amplitude vector and k=k 1,...,k d T R d the spatial wave number, which satisfies the following eigenvalue problem ωb= d δ k j A0 j α j + ν 2 β +V 0 I 4 B Solving the above equation, we can get the dispersion relation of the Dirac equation 1.7 ω := ωk= V 0 ± 1 2 ν δ k A 0 2, k R d In 2D and 1D, i.e. d = 2 or 1 in 1.7, similar as those in [8], the Dirac equation 1.7 can be decoupled into two simplified PDEs with Φ := Φt,x =φ 1 t,x,φ 2 t,x T C 2 satisfying iδ t Φ = i δ d σ j j + ν 2 σ 3 Φ+ VxI 2 d A j xσ j Φ, x R d, 1.17

5 A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation 5 where Φ =ψ 1,ψ 4 T or Φ =ψ 2,ψ 3 T. Again, to study the dynamics of the Dirac equation 1.17, the initial condition is usually taken as Φt = 0,x=Φ 0 x, x R d Similarly, the Dirac equation 1.17 with 1.18 is dispersive, time-symmetric, and it conserves the total probability [9] and the energy [9] EΦt, := R d Φt, 2 := dx= R d Φt,x 2 i δ d 2 R d φ j t,x 2 dx Φ0, 2 = Φ 0 2, t 0, 1.19 Φ σ j j Φ+ ν 2 Φ σ 3 Φ+Vx Φ 2 d A j xφ σ j Φ EΦ 0, t Again, introduce the total probability density ρ := ρt, x as ρt,x= 2 ρ j t,x=φt,x Φt,x, x R d, 1.21 where the probability density ρ j := ρ j t,x of the j-th,2 component is defined as and the current density Jt,x=J 1 t,x,...,j d t,x T as ρ j t,x= φ j t,x 2, x R d, 1.22 dx J l t,x= 1 Φt,x σ l Φt,x, l = 1,...,d, 1.23 then the same conservation law 1.14 can be obtained from the Dirac equation 1.17 [9]. Similarly, if the electric potential V is perturbed by a real constant V 0, i.e., V V + V 0, then the solution Φt,x e i V 0 t δ Φt, x, which implies that the probability density of each component ρ j j = 1,2 and the total probability density ρ are all unchanged. In addition, when d = 1, if the magnetic potential A 1 is perturbed by a real constant A 0 1, i.e., A 1 A 1 + A 0 1, then the solution A0 Φt,x ei 1 t δ σ 1 Φt,x, which implies that only the total probability density ρ is unchanged; however, this property is unfortunately not valid in 2D. Furthermore, if the external electromagnetic potentials are all real constants, i.e. Vx V 0 and A j x A 0 j j = 1,...,d with A0 =A 0 1,...,A0 d T, the Dirac equation 1.17 admits the plane wave solution Φt,x=Be ik x ω δ t with ω the time frequency, B R 2 the amplitude vector and k = k 1,...,k d T R d the spatial wave number, which satisfies the following

6 6 Weizhu Bao, Jia Yin Standard Dirac Eq. Weyl Eq. = δ = 1 ν 0 m 0 massless limit = δ = ν = 1 = ν = 1 Dirac Eq. 1.7 or 1.17 with, δ, ν δ 0 h 0 classical regime semiclasscial limit δ = ν = 1 0 c nonrelativistic limit Schrödinger Eq. or Pauli Eq. relativistic Euler Eqs. ν = 1 δ 0 h 0 semiclasscial limit ν = 1 0 c nonrelativistic limit Euler Eqs. Fig. 1.1 Diagram of different parameter regimes and limits of the Dirac equation 1.7 or eigenvalue problem ωb= d δ k j A0 j σ j + ν 2 σ 3+V 0 I 2 B Solving the above equation, we can get the dispersion relation of the Dirac equation 1.17 ω := ωk= V 0 ± 1 2 ν δ k A 0 2, k R d If one sets the mass unit m s = m, length unit x s = h mc, and time unit t s = x s c = h mc 2, then = δ = ν = 1, which corresponds to the classical or standard scaling. This choice of x s, m s and t s is appropriate when the wave speed is at the same order of the speed of light. However, a different choice of x s, m s and t s is more appropriate when the wave speed is much smaller than the speed of light. We remark here that the choice of x s, m s and t s determines the observation scale of time evolution of the system and decides which phenomena can be resolved by discretization on specified spatial/temporal grids and which phenomena is visible by asymptotic analysis. Different parameter regimes could be considered for the Dirac equation 1.7 or 1.17, which are displayed in Fig. 1.1: Standard or classical regime, i.e. = δ = ν = 1 m s = m, x s = h mc, and t s = h mc 2, the wave speed is at the order of the speed of light. In this parameter regime, formally the

7 A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation 7 dispersion relation 1.16 or 1.25 suggests ωk=o1 when k =O1 and thus the solution propagates waves with wavelength at O1 in space and time. In addition, if the initial data Ψ 0 = O1 in 1.8 or Φ 0 = O1 in 1.18, then the solution Ψ = O1 of 1.7 with 1.8 or Φ = O1 of 1.17 with 1.18, which implies that the probability density ρ = O1 in 1.11 or 1.21, current density J = O1 in 1.13 or 1.23 and the energy EΨt, = O1 in 1.10 or EΦt, = O1 in There were extensive analytical and numerical studies for the Dirac equation 1.7 or 1.17 with = δ = ν = 1 in the literatures. For the existence and multiplicity of bound states and/or standing wave solutions, we refer to [26, 27, 32, 40, 41, 52] and references therein. In this parameter regime, for the numerical part, many efficient and accurate numerical methods have been proposed and analyzed [3], such as the finite difference time domain FDTD methods [4, 50], time-splitting Fourier pseudospectral TSFP method [9, 42], exponential wave integrator Fourier pseudospectral EWI-FP method [9], the Gaussian beam method [62], etc. Massless limit regime, i.e. = δ = 1 and 0<ν 1 x s = h m s c and t s = h m s c 2, the mass of the particle is much less than the mass unit. In this parameter regime, the Dirac equation 1.7 or 1.17 converges regularly to the Weyl equation [51, 63] with linear convergence rate in terms of ν. Any numerical methods for the Dirac equation 1.7 or 1.17 in the standard regime can be applied in this parameter regime. Nonrelativistic limit regime, i.e. δ = ν= 1 and 0< 1 m s = m and t s = mx2 s h, i.e. the wave speed is much less than the speed of light. In this parameter regime, formally the dispersion relation 1.16 or 1.25 suggests ωk= 2 + O1 when k =O1 and thus the solution propagates waves with wavelength at O 2 and O1 in time and space, respectively, when 0 < 1. In addition, if the initial data Ψ 0 = O1 in 1.8 or Φ 0 = O1 in 1.18, then the solution Ψ = O1 of 1.7 with 1.8 or Φ = O1 of 1.17 with 1.18, which implies that the probability density ρ = O1 in 1.11 or 1.21, current density J = O 1 in 1.13 or 1.23 and the energy EΨt, = O 2 in 1.10 or EΦt, =O 2 in The highly oscillatory nature of the solution in time and the unboundedness of the energy bring significant difficulty in mathematical analysis and numerical simulation of the Dirac equation in the nonrelativistic regime, i.e. 0 < 1. By diagonalizing the Dirac operator and using proper ansatz, one can show that the Dirac equation 1.7 or 1.17 converges singularly to the Pauli equation [16, 43] and/or the Schrödinger equation [6, 16] when 0 +. Rigorous error estimates were established for the FDTD, TSFP and EWI-FP methods in this parameter regime [9], which depend explicitly on the mesh size h, time step τ and the small parameter. Recently, a uniformly accurate multiscale time integrator pseudospectral method was proposed and analyzed for the Dirac equation in the nonrelativistic limit regime, which converges uniformly with respect to 0, 1] [8, 46].

8 8 Weizhu Bao, Jia Yin Semiclassical limit regime, i.e. = ν = 1 and 0 < δ 1 m s = m and t s = x s c, the quantum effect could be neglected. In this parameter regime, the solution propagates waves with wavelength at Oδ in space and time [19] when 0<δ 1. In addition, if the initial data Ψ 0 = O1 in 1.8 or Φ 0 = O1 in 1.18, then the solution Ψ = O1 of 1.7 with 1.8 or Φ = O1 of 1.17 with 1.18, which implies that the probability density ρ = O1 in 1.11 or 1.21, current density J = O1 in 1.13 or 1.23 and the energy EΨt, = O1 in 1.10 or EΦt, = O1 in The highly oscillatory nature of the solution in time and space brings significant difficulty in mathematical analysis and numerical simulation of the Dirac equation in the semiclassical limit regime, i.e. 0 < δ 1. By using the Wigner transformation method, one can show that the Dirac equation 1.7 or 1.17 converges singularly to the relativistic Euler equations [5, 39, 53]. Similar to the analysis of different numerical methods for the Schrödinger equation in the semiclassical limit regime [2,7,12,13,21,22,45], it is an interesting question to establish rigorous error bounds of different numerical methods for the Dirac equation in the semiclassical limit regime such that they depend explicitly on mesh size h, time step τ as well as the small parameter δ 0,1]. Simultaneously nonrelativistic and massless limit regimes, i.e. δ = 1, ν and 0 < 1 t s = m sx 2 s h, the wave speed is much less than the speed of light and the mass of the particle is much less than the mass unit. Here we assume ν = ν 0 with ν 0 > 0 a constant independent of 0, 1]. In this case, the Dirac equation 1.7 can be re-written as d = 1,2,3 i t Ψ = i 1 d α j j + ν 0 β Ψ + VxI 4 d A j xα j Ψ, x R d, 1.26 and respectively, the Dirac equation 1.17 can be re-written as d = 1,2 i t Φ = i 1 d σ j j + ν 0 σ 3 Φ+ VxI 2 d A j xσ j Φ, x R d In this parameter regime, formally the dispersion relation 1.16 or 1.25 suggests ωk = O 1 when k = O1 and thus the solution propagates waves with wavelength at O and O1 in time and space, respectively, when 0 < 1. In addition, if the initial data Ψ 0 = O1 in 1.8 or Φ 0 = O1 in 1.18, then the solution Ψ = O1 of 1.26 with 1.8 or Φ = O1 of 1.27 with 1.18, which implies that the probability density ρ = O1 in 1.11 or 1.21, current density J = O 1 in 1.13 or 1.23 and the energy EΨt, =O 1 in 1.10 or EΦt, =O 1 in Again, the highly oscillatory nature of the solution in time and the unboundedness of the energy bring significant difficulty in mathematical analysis and numerical simulation of the Dirac equation in this parameter regime. In fact, it is an interesting question to study the singular limit of the Dirac equation 1.26 or 1.27 when 0 + and establish

9 A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation 9 rigorous error bounds of different numerical methods for the Dirac equation in this parameter regime such that they depend explicitly on mesh size h, time step τ as well as the small parameter 0, 1]. First-order and second-order in time time-splitting spectral methods have been proposed and analyzed for the Dirac equation 1.7 or 1.17 [9]. Extension to higher order, e.g. fourth-order, time-splitting spectral methods can be done straightforward by adapting the high order splitting methods [15,47,57], e.g. the standard fourth-order splitting S 4 [37, 55,64] or the fourth-order partitioned Runge-Kutta S 4RK splitting method [17,38]. As it was observed in the literature [47], the S 4 splitting method has to use negative time step in at least one of the sub-problems at each time interval [37,55,64], which causes some kind of drawbacks in practical computation, and the number of sub-problems in the S 4RK splitting method at each time interval is much bigger than that of the S 4 splitting method [17], which increases the computational cost at each time step a lot. Motivated by the fourthorder gradient symplectic integrator for the Schödinger equation invented by [23 25], a new fourth-order compact time-splitting S 4c Fourier pseudospectral method will be proposed for the Dirac equation by splitting the Dirac equation into two parts together with using the double commutator between them to integrate the Dirac equation at each time interval. The method is explicit, fourth-order in time and spectral order in space. We compare the accuracy and efficiency as well as long time behavior of the S 4c method with many other existing time-splitting methods for the Dirac equation. Numerical results demonstrate the advantage of the proposed S 4c in terms of efficiency and accuracy, especially in 1D and high dimensions 2D and 3D without magnetic potential. We also report the spatial/temporal resolution of the S 4c method for the Dirac equation in different parameter regimes. The rest of the paper is organized as follows. In section 2, we review different timesplitting schemes for differential equations. In section 3, we calculate the double commutator between the two parts decoupled from the Dirac equation. A fourth-order compact time-splitting Fourier pseudospectral method is proposed for the Dirac equation in section 4. In section 5, we compare accuracy and efficiency as well as long time behavior of different time-splitting methods for the Dirac equation. In section 6, we report spatial/temporal resolution of the fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation in different parameter regimes. Finally, some concluding remarks are drawn in section 7. Throughout the paper, we adopt the standard Sobolev spaces and the corresponding norms and adopt A B to mean that there exists a generic constant C > 0 independent of, τ, h, δ and ν such that A C B. 2 Review of different time-splitting schemes Splitting or split-step or time-splitting methods have been widely used in numerically integrating differential equations [47]. Combined with different spatial discretization schemes,

10 10 Weizhu Bao, Jia Yin they have also been applied in solving partial differential equations [47]. For details, we refer to [56 58] and references therein. For simplicity of notations and the convenience of readers, here we review several timesplitting schemes for integrating a differential equation in the form t ut,x=t +Wut,x, 2.1 with the initial data u0,x=u 0 x, 2.2 where T and W are two time-independent operators. For any time step τ > 0, formally the solution of 2.1 with 2.2 can be represented as uτ,x=e τt+w u 0 x. 2.3 A splitting or split-step or time-splitting scheme can be designed by approximating the operator e τt+w by a product of a sequence of e τt and e τw [55,64], i.e. e τt+w Π n ea jτ T e b jτ W, 2.4 where n 1, a j R and b j R j = 1,...,n are to be determined such that the approximation has certain order of accuracy in terms of τ [55,64]. Without loss of generality, here we suppose that the computation for e τw is easier and/or more efficient than that for e τt. 2.1 First-order and second-order time-splitting methods Taking n=1and a 1 = b 1 = 1 in 2.4, one can obtain the first-order Lie-Trotter splitting S 1 method as [61] uτ,x S 1 τu 0 x := e τt e τw u 0 x. 2.5 In this method, one needs to integrate the operator T once and the operator W once. By using Taylor expansion, one can formally show the local truncation error as [54] uτ,x S 1 τu 0 x C 1 τ 2, 2.6 where C 1 > 0 is a constant independent of τ and is a norm depending on the problem. Thus the method is formally a first-order integrator [47]. Similarly, taking n=2, a 1 = 0, b 1 = 1 2, a 2 = 1 and b 2 = 1 2, one can obtain the secondorder Strang splitting S 2 method as [54] uτ,x S 2 τu 0 x := e τ 2 W e τt e τ 2 W u 0 x. 2.7

11 A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation 11 In this method, one needs to integrate the operator T once and the operator W twice. Again, by using Taylor expansion, one can formally show the local truncation error as [54] uτ,x S 2 τu 0 x C 2 τ 3, 2.8 where C 2 > 0 is a constant independent of τ. Thus it is formally a second-order integrator [47]. 2.2 Fourth-order time-splitting methods High order, especially fourth-order, splitting methods for 2.1 with 2.2 via the construction 2.4 had been extensively studied in the literature [23,24]. For simplicity, here we only mention a popular fourth-order Forest-Ruth or Yoshida splitting S 4 method [37,55,64] as uτ,x S 4 τu 0 x := S 2 w 1 τs 2 w 2 τs 2 w 1 τu 0 x, 2.9 where 1 w 1 = 2 2 1/3, w 2 = 21/3 2 21/ In this method, one needs to integrate the operator T three times and the operator W four times. Again, by using Taylor expansion, one can formally show the local truncation error as [37] uτ,x S 4 τu 0 x C 4 τ 5, 2.11 where C 4 > 0 is a constant independent of τ. Thus it is formally a fourth-order integrator [47]. Due to that negative time steps, e.g. w 2 < 0, are used in the method, in general, it cannot be applied to solve dissipative differential equations. In addition, as it was noticed in the literature [47], some drawbacks of the S 4 method were reported, such as the constant C 4 is usually much larger than C 1 and C 2, and the fourth-order accuracy could be observed only when τ is very small [47,58]. To overcome the drawbacks of the S 4 method, the fourth-order partitioned Runge- Kutta splitting S 4RK method was proposed [17,38] as uτ,x S 4RK τu 0 x 2.12 := e a 1τW e b 1τT e a 2τW e b 2τT e a 3τW e b 3τT e a 4τW e b 3τT e a 3τW e b 2τT e a 2τW e b 1τT e a 1τW u 0 x,

12 12 Weizhu Bao, Jia Yin where a 1 = , a 2 = , a 3 = , a 4 = 1 2a 1 + a 2 + a 3, b 1 = , b 2 = , b 3 = 1 2 b 1+ b 2. In this method, one needs to integrate the operator T six times and the operator W seven times. Again, by using Taylor expansion, one can formally show the local truncation error as [17] uτ,x S 4RK τu 0 x C 4 τ 5, 2.13 where C 4 > 0 is a constant independent of τ. Thus it is formally a fourth-order integrator [47]. It is easy to see that the computational cost of the S 4RK method is about two times that of the S 4 method. In this method, negative time steps, e.g. a 3 < 0, have also been used. 2.3 Fourth-order compact time-splitting methods To avoid the negative time steps and motivated by the numerical integration of the Schrödinger equation [23 25], a fourth-order gradient symplectic integrator was proposed by S. A. Chin [23] as uτ,x S 4c τu 0 x := e 6 1 τw e 1 2 τt e 3 2 τŵ e 1 2 τt e 6 1 τw u 0 x, 2.14 where Ŵ := W τ 2 [W,[T,W]], 2.15 with [T,W] := TW WT the commutator of the two operators T and W and [W,[T,W]] a double commutator. Again, by using Taylor expansion, one can formally show the local truncation error as [23,24] uτ,x S 4c τu 0 x Ĉ 4 τ 5, 2.16 where Ĉ4 > 0 is a constant independent of τ. Thus it is formally a fourth-order integrator [47]. In this method, in general, one needs to integrate the operator T twice and the operator W three times under the assumption that the computation of Ŵ is equivalent to that of W. Thus it is more efficient than the S 4 and S 4RK methods. In this sense, it is more appropriate to name it as a fourth-order compact splitting S 4c method since, at each time step, the number of sub-steps in it is much less than those in the S 4 and S 4RK methods. Another advantage of the S 4c method is that there is no negative time step in it.

13 A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation 13 S 1 S 2 S 4 S 4RK S 4c T W Table 2.1 The numbers of operators T and W to be implemented in different time-splitting methods. For comparison, Table 2.1 lists the numbers of T and W to be integrated by different splitting methods. From it, under the assumption that the computation for e τw is easier and/or more efficient than that for e τt and the computation of e τŵ is similar to that for e τw, we could draw the following conclusions: i the computational time of S 2 is almost the same as that of S 1 ; ii the computational time of S 4c is about two times of that of S 2 or S 1 ; iii among the three fourth-order splitting methods, S 4c is the most efficient and S 4RK is the most expensive. 3 Computation for the double commutator [W,[T,W]] In this section, we first show that the double commutator [W,[T,W]] is linear in T and then compute it for the Dirac equations 1.17 for d = 1,2 and 1.7 for d = 1,2,3. Lemma 1 Let T and W be two operators, then we have [W,[T,W]]=2W TW WW T TWW. 3.1 Thus the double commutator [W,[T,W]] is linear in T, i.e. for any two operators T 1 and T 2, we have [W,[a 1 T 1 + a 2 T 2,W]]=a 1 [W,[T 1,W]]+a 2 [W,[T 2,W]], a 1,a 2 R. 3.2 Proof Noticing[T,W] := TW W T, we have [W,[T,W]] = [W,TW WT]= WTW W T TW WTW = W TW WWT TWW +WTW = 2W TW WWT TWW. 3.3 From 3.3, it is easy to see that the double commutator [W,[T,W]] is linear in T, i.e. 3.2 is valid.

14 14 Weizhu Bao, Jia Yin 3.1 Double commutators of the Dirac equation in 1D Lemma 2 For the Dirac equation 1.17 in 1D, i.e. d = 1, define we have T = 1 σ 1 1 iν δ 2 σ 3, Proof Combining 3.4 and 3.2, we obtain Noticing 3.1 and 3.4, we have W = i δ VxI 2 A 1 xσ 1, 3.4 [W,[T,W]]= 4iν δ 3 2 A2 1xσ [W,[T,W]]= 1 [W,[σ 1 1,W]] iν δ 2[W,[σ 3,W]]. 3.6 [W,[σ 1 1,W]] = 2 i VxI2 A 1 xσ 1 σ 1 1 i VxI2 A 1 xσ 1 δ δ i 2 VxI2 A 1 xσ 1 σ 1 1 σ 1 1 i 2 VxI2 A 1 xσ 1 δ δ = 2 VxI2 δ 2 A 1 xσ 1 σ1 1 VxI2 A 1 xσ σ1 VxI2 δ 2 A 1 xσ δ 2 σ VxI2 A 1 xσ 1 = 2 δ 2 σ 1 VxI2 A 1 xσ 1 1 VxI2 A 1 xσ 1 2 δ 2 σ VxI2 A 1 xσ δ 2 σ VxI2 A 1 xσ δ 2 σ 1 VxI2 A 1 xσ 1 1 VxI2 A 1 xσ 1 = [W,[σ 3,W]] = 2 i VxI2 A 1 xσ 1 σ 3 i VxI2 A 1 xσ 1 δ δ i 2 VxI2 A 1 xσ 1 σ 3 σ 3 i 2 VxI2 A 1 xσ 1 δ δ = 2 VxI2 δ 2 A 1 xσ 1 VxI2 + A 1 xσ 1 σ σ3 VxI2 δ 2 A 1 xσ σ3 VxI2 δ 2 + A 1 xσ 1 = 1 δ 2 2V 2 xi 2 2A 2 1xI 2 V 2 xi 2 + A 2 1xI 2 2A 1 xvxσ 1 V 2 xi 2 + A 2 1 xi 2+ 2A 1 xvxσ 1 σ 3 = 1 δ 2 4A 2 1 xi 2 σ3 = 4 δ 2 A2 1xσ

15 A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation 15 Plugging 3.7 and 3.8 into 3.6, we can obtain 3.5 immediately. Combining 3.5, 3.4 and 2.15, we have Ŵ = W τ 2 [W,[T,W]]= i VxI 2 A 1 xσ 1 iντ2 δ 12δ 3 2 A2 1 xσ 3, 3.9 which immediately implies that the computation of e τŵ is similar or at almost the same computational cost to that for e τw in this case. Corollary 1 For the Dirac equation 1.7 in 1D, i.e. d = 1, define T = 1 α 1 1 iν δ 2 β, W = i δ VxI 4 A 1 xα 1, 3.10 we have [W,[T,W]]= 4iν δ 3 2 A2 1 xβ Double commutators of the Dirac equation in 2D and 3D Similar to the 1D case, we have see detailed computation in Appendix A Lemma 3 For the Dirac equation 1.17 in 2D, i.e. d = 2, define T = 1 σ σ 2 2 iν δ 2 σ 3, W = i δ VxI 2 A 1 xσ 1 A 2 xσ 2, 3.12 we have [W,[T,W]]=F 3 x+f 1 x 1 + F 2 x 2, 3.13 where F 1 x = 4 A 2 δ 2 2 xσ 1+ A 1 xa 2 xσ 2, F 2 x= 4 δ 2 A 1 xa 2 xσ 1 A 2 1 xσ 2, F 3 x = 4 δ 2 A 1 x 2 A 2 x A 2 x 1 A 2 x σ δ 2 A 2 x 1 A 1 x A 1 x 2 A 1 x + 4i δ 2 A 2 x 1 Vx A 1 x 2 Vx ν A 2 δ 1 x+a 2 2x σ 3. Corollary 2 For the Dirac equation 1.7 in 2D, i.e. d = 2, define T = 1 α α 2 2 iν δ 2 β, W = i VxI 2 A 1 xα 1 A 2 xα 2, 3.14 δ σ 2 we have [W,[T,W]]=F 3 x+f 1 x 1 + F 2 x 2, 3.15

16 16 Weizhu Bao, Jia Yin where F 1 x = 4 A 2 δ 2 2 xα 1+ A 1 xa 2 xα 2, F 2 x= 4 δ 2 A 1 xa 2 xα 1 A 2 1 xα 2, F 3 x = 4 δ 2 A 1 x 2 A 2 x A 2 x 1 A 2 x α δ 2 A 2 x 1 A 1 x A 1 x 2 A 1 x + 4i δ 2 A 2 x 1 Vx A 1 x 2 Vx γα 3 4iν A 2 δ x+a 2 2x β, α 2 where 0 I 2 γ = I 2 0 For the Dirac equation 1.7 in 3D, i.e. d = 3, we have see detailed computation in Appendix B Lemma 4 For the Dirac equation 1.7 in 3D, i.e. d = 3, define T = 1 3 α j j iν δ 2 β, W = i VxI 4 δ 3 A j xα j, 3.17 we have [W,[T,W]]=F 4 x+f 1 x 1 + F 2 x 2 + F 3 x 3, 3.18 where F 1 x = 4 δ 2 A 2 2 x+a2 3 x α 1 + A 1 xa 2 xα 2 + A 1 xa 3 xα 3, F 2 x = 4 A δ 2 2 xa 1 xα 1 A 2 1 x+a2 3 x α 2 + A 2 xa 3 xα 3, F 3 x = 4 A δ 2 3 xa 1 xα 1 + A 3 xa 2 xα 2 A 2 1x+A 2 2x α 3, F 4 x = 4 δ 2 A 1 x 2 A 2 x+ 3 A 3 x A 2 x 1 A 2 x A 3 x 1 A 3 x α δ 2 A 2 x 1 A 1 x+ 3 A 3 x A 1 x 2 A 1 x A 3 x 2 A 3 x α δ 2 A 3 x 1 A 1 x+ 2 A 2 x A 1 x 3 A 1 x A 2 x 3 A 2 x α 3 + 4i A δ 2 1 x 2 A 3 x 3 A 2 x + A 2 x 3 A 1 x 1 A 3 x +A 3 x 1 A 2 x 2 A 1 x γ+ 4i δ 2 A 3 x 2 Vx A 2 x 3 Vx γα 1 + 4i δ 2 A 1 x 3 Vx A 3 x 1 Vx γα 2 + 4i δ 2 A 2 x 1 Vx A 1 x 2 Vx γα 3 4iν δ 3 2 A 2 1x+A 2 2x+A 2 3x β. From Lemmas 2, 3 and 4 and Corollaries 1 and 2, it is easy to observe that the double commutator will vanish when the Dirac equation 1.17 or 1.7 has no magnetic potentials.

17 A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation 17 Lemma 5 For the Dirac equation 1.17 in 1D and 2D, and for the Dirac equation 1.7 in 1D, 2D and 3D, when there is no magnetic potential, i.e., when A 1 x=a 2 x=a 3 x 0, we have [W,[T,W]]= A fourth-order compact time-splitting Fourier pseudospectral method In this section, we present a fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation 1.7 or 1.17 by using the S 4c method 2.14 for time integration followed by the Fourier pseudospectral method for spatial discretization. 4.1 Time integration by the S 4c method in 1D For simplicity of notations, we present the numerical method for 1.17 in 1D first. Similar to most works in the literatures for the analysis and computation of the Dirac equation cf. [8 10, 14] and references therein, in practical computation, we truncate the whole space problem onto an interval Ω = a, b with periodic boundary conditions. The truncated interval is large enough such that the truncation error is negligible. In 1D, the Dirac equation 1.17 with periodic boundary conditions collapses to iδ t Φ = i δ σ 1 x + ν 2 σ 3 Φ+ VxI 2 A 1 xσ 1 Φ, x Ω, t > 0, Φt,a=Φt,b, x Φt,a= x Φt,b, t 0; 4.1 Φ0,x=Φ 0 x, a x b; where Φ := Φt,x, Φ 0 a=φ 0 b and Φ 0 a=φ 0 b. Choose a time step τ > 0, denote t n = nτ for n 0 and let Φ n x be an approximation of Φt n,x. Re-writing the Dirac equation 4.1 as t Φ = 1 σ 1 x iν δ 2 σ 3 Φ i VxI 2 A 1 xσ 1 Φ :=T +WΦ, 4.2 δ then we can apply the S 4c method 2.14 for time integration over the time interval [t n,t n+1 ] as Φ n+1 x=s 4c τφ n x := e 1 6 τw e 1 2 τt e 2 3 τŵ e 1 2 τt e 1 6 τw Φ n x, a x b, n 0, 4.3 where the two operators T and W are given in 3.4 and the operator Ŵ is given in 3.9. In order to calculate e 1 2 τt, we can discretize it in space via Fourier spectral method and then integrate in phase space or Fourier space in time exactly [9, 14]. Since W is diagonalizable

18 18 Weizhu Bao, Jia Yin [9], e 1 6 τw can be evaluated very efficiently [9]. For e 2 3 τŵ, plugging 1.3 into 3.9, we can diagonalize it as Ŵ = i VxI 2 A 1 xσ 1 iντ2 δ 12δ 3 2 A2 1xσ 3 = ip 2 xλ 2 xp 2 x := Ŵx, 4.4 where Λ 2 x=diagλ 2 + x,λ 2 x with λ 2 ± x= Vx δ ± A 1x 144δ ν 2 τ 4 A 2 1 x and 1 β1 x+β 2 x β1 x β 2 x P 2 x= 2β1 x β 1 x β 2 x, a x b, 4.5 β 1 x+β 2 x 12δ 3 2 with β 1 x= 144δ ν 2 τ 4 A 2 1 x, β 2x=ντ 2 A 1 x, a x b. 4.6 Thus we have e 2 3 τŵ = e 2i 3 τp 2xΛ 2 xp 2 x = P 2 xe 2i 3 τλ 2x P 2 x, a x b Full discretization in 1D Choose a mesh size h := x = b a M with M being an even positive integer and denote the grid points as x j := a+ jh, for j=0,1,...,m. Denote X M ={U =U 0,U 1,...,U M T U j C 2, j=0,1,...,m, U 0 = U M }. For any U X M, we denote its Fourier representation as U j = M/2 1 l= M/2 Ũ l e iµ lx j a = where µ l and Ũ l C 2 are defined as M/2 1 l= M/2 Ũ l e 2i jlπ/m, j=0,1,...,m, 4.8 µ l = 2lπ b a, Ũ l = 1 M M 1 j=0 U j e 2i jlπ/m, l = M 2,..., M For U X M and ux L 2 Ω, their l 2 -norms are defined as U 2 M 1 l 2 := h j=0 U j 2, u 2 M 1 l 2 := h j=0 ux j Let Φ n j be the numerical approximation of Φt n,x j and denote Φ n = Φ n 0,Φn 1,...,Φn M T X M as the solution vector at t = t n. Take Φ 0 j = Φ 0 x j for j= 0,...,M, then a fourth-order compact time-splitting Fourier pseudospectral S 4c discretization for the Dirac equation

19 A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation is given as Φ 1 j = e τ 6 Wx j Φ n j = P 1 e iτ 6 Λ 1x j P 1 Φ n j, Φ 2 M/2 1 τγ j = e l Φ1 l= M/2 M/2 1 l eiµ lx j a = l= M/2 Q l e iτd l Q l Φ1 l e2i jlπ/m, Φ 3 j = e 2τ 3 Ŵx j Φ 2 j = P 2 x j e 2iτ 3 Λ 2x j P 2 x j Φ 2 j, j=0,1,...,m, Φ 4 M/2 1 j = l= M/2 τγ e l Φ3 M/2 1 l eiµ lx j a = l= M/2 Φ n+1 j = e τ 6 Wx j Φ 4 j = P 1 e iτ 6 Λ 1x j P 1 Φ 4 j, Q l e iτd l Q l Φ3 l e2i jlπ/m, 4.11 where Wx j := i Vx j I 2 A 1 x j σ 1 = ip 1 Λ 1 x j P δ 1, j = 0,1,...,M, Γ l = iµ l σ 1 iν δ 2 σ 3 = iq l D l Q l, l = M 2,..., M , with D l = diag 1 δ 2 with λ 1 ± x= 1 δ Vx±A1 x, η l = P 1 =, Q l = ν 2 + δ 2 2 µ l ν 2, 1 δ 2 + δ 2 2 µ 2 2 l ν 2 + δ 2 2 µ l 2, and 1 2ηl η l + ν, Λ 1 x=diag λ 1 + x,λ 1 x ηl + ν δ µ l, l = M δ µ l η l + ν 2,..., M We remark here that full discretization by other time-splitting methods together with Fourier pseudospectral method for spatial discretization can be implemented similarly [9] and the details are omitted here for brevity. 4.3 Mass conservation in 1D The S 4c method 4.11 is explicit, its memory cost is OM and its computational cost per time step is OM lnm, it is fourth-order accurate in time and spectral accurate in space. In addition, it conserves the total probability in the discretized level, as shown in the following lemma. Lemma 6 For any τ > 0, the S 4c method 4.11 conserves the mass in the discretized level, i.e. Φ n+1 2 M 1 := h l 2 Φ n+1 j j=0 2 h M 1 j=0 Φ 0 j 2 = h M 1 j=0 Φ 0 x j 2 2 = Φ 0 2, n l

20 20 Weizhu Bao, Jia Yin Proof Noticing Wx j = Wx j and thus e 6 τ Wx τ j e 6 Wx j = I 2, from 4.11 and summing for j=0,1,...,m 1, we get Φ n+1 2 M 1 = h l 2 Φ n+1 j j=0 M 1 = h j=0 Similarly, we have 2 = h M 1 j=0 Φ 4 j I 2 Φ 4 j = h e 6 τ Wx j Φ 4 2 = h M 1 j=0 j Φ 4 j M 1 j=0 Φ 4 j e 6 τ Wx τ j e 6 Wx j Φ 4 j 2 = Φ 4 2 2, n Φ 3 2 Φ = 2 2 Φ l 2 l 2, 1 2 Φ = n 2 l 2 l2, n Similarly, using the Parsval s identity and noticing Γl = Γ l and thus e τγ l e τγ l = I 2, we get Φ 4 2 Φ = 3 2 Φ l 2 l 2, 2 2 Φ = 1 2 l 2 l Combining 4.15, 4.16 and 4.17, we obtain Φ n+1 2 Φ = 4 2 Φ = 3 2 Φ = 2 2 Φ = 1 2 Φ = n 2 l 2 l 2 l 2 l 2 l 2 l2, n Using the mathematical induction, we get the mass conservation l 4.4 Discussion on extension to 2D and 3D When there is no magnetic potential, i.e., when A 1 x = A 2 x = A 3 x 0 in the Dirac equation 1.17 in 2D and 1.7 in 2D and 3D, from Lemma 5, we know that the double commutator [W,[T,W]]=0. In this case, noting 2.15, we have Ŵ = W τ 2 [W,[T,W]]= W Then the S 4c method 2.14 collapses to uτ,x S 4c τu 0 x := e 6 1 τw e 1 2 τt e 3 2 τw e 1 2 τt e 6 1 τw u 0 x Applying the S 4c method 4.20 to integrate the Dirac equation 1.17 in 2D over the time interval [t n,t n+1 ] with Φt n,x=φ n x given, we obtain Φ n+1 x=s 4c τφ n x=e 1 6 τw e 1 2 τt e 3 2 τw e 1 2 τt e 6 1 τw Φ n x, x Ω, n 0, 4.21 where T and W are given in Similarly, applying the S 4c method 4.20 to integrate the Dirac equation 1.7 in 2D and 3D over the time interval [t n,t n+1 ] with Ψt n,x= Ψ n x

21 A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation 21 given, we obtain Ψ n+1 x=s 4c τψ n x=e 1 6 τw e 1 2 τt e 2 3 τw e 1 2 τt e 1 6 τw Ψ n x, x Ω, n 0, 4.22 where T and W are given in 3.14 and 3.17 for 2D and 3D, respectively. In practical computation, the operators e 1 6 τw and e 2 3 τw in 4.21 and 4.22 can be evaluated in physical space directly and easily [9]. For the operator e 1 2 τt, it can be discretized in space via Fourier spectral method and then integrate in phase space or Fourier space in time exactly. For details, we refer to [9,14] and references therein. In fact, the implementation of the S 4c method in this case is much simpler than that of the S 4 and S 4RK methods. Of course, when the magnetic potential is nonzero in the Dirac equation 1.17 in 2D and 1.7 in 2D and 3D, one has to adapt the formulation 4.20 for S 4c method. In this case, the main difficulty is how to efficiently and accurately evaluate the operator e 2 3 τŵ. This can be done by using the method of characteristics and the nonuniform fast Fourier transform NUFFT, which has been developed for the magnetic Schrödinger equation. For details, we refer to [20,44] and references therein. Of course, it is a little more tedious in practical implementation for S 4c method than that for the S 4 and S 4RK methods in this situation. 5 Comparision of different time-splitting methods In this section, we compare the fourth-order compact time-splitting Fourier pseudospectral S 4c method 4.11 with other time-splitting methods including the first-order time-splitting S 1 method, the second-order time-splitting S 2 method, the fourth-order time-splitting S 4 method and the fourth-order partitioned Runge-Kutta time-splitting S 4RK method in terms of accuracy and efficiency as well as long time behavior. 5.1 An example in 1D For simplicity, we first consider an example in 1D. In the Dirac equation 1.17, we take d = 1, = δ = ν = 1 and Vx= 1 x 1+x 2, A 1x= x+12, x R x2 The initial data in 1.18 is taken as: φ 1 0,x=e x2 /2, φ 2 0,x=e x 12 /2, x R. 5.2 The problem is solved numerically on a bounded domain Ω = 32,32, i.e. a= 32 and b=32.

22 22 Weizhu Bao, Jia Yin h 0 = 1 h 0 /2 h 0 /2 2 h 0 /2 3 S E E-5 S E E E-10 S E E E-10 S 4c E E E-10 S 4RK E E E-10 Table 5.1 Spatial errors e Φ t = 6 of different time-splitting methods under different mesh size h for the Dirac equation 1.17 in 1D. Due to the fact that the exact solution is not available, we obtain a numerical exact solution by using the S 4c method with a fine mesh size h e = 1 16 and a small time step τ e = Let Φ n be the numerical solution obtained by a numerical method with mesh size h and time step τ. Then the error is quantified as e Φ t n = Φ n M 1 Φt n, l 2 = h Φt n,x j Φ n j In order to compare the spatial errors, we take time step τ = τ e = 10 5 such that the temporal discretization error could be negligible. Table 5.1 lists numerical errors e Φ t = 6 for different time-splitting methods under different mesh size h. We remark here that, for the S 1 method, in order to observe the spatial error when the mesh size h=h 0 /2 3, one has to choose time step τ which is too small and thus the error is not shown in the table for this case. From Table 5.1, we could see that all the numerical methods are spectral order accurate in space cf. each row in Table 5.1. In order to compare the temporal errors, we take mesh size h=h e = 16 1 such that the spatial discretization error could be negligible. Table 5.2 lists numerical errors e Φ t = 6 for different time-splitting methods under different time step τ. In the table, we use second s as the unit for CPU time. For comparison, Figure 5.1 plots e Φ t = 6 and e Φ t = 6/τ α with α taken as the order of accuracy of a certain numerical method in order to show the constants C 1 in 2.6, C 2 in 2.8, C 4 in 2.11, C 4 in 2.13 and Ĉ 4 in 2.16 for different time-splitting methods under different time step τ. From Table 5.2 and Figure 5.1, we can draw the following conclusions: i The S 1 method is first-order in time, the S 2 method is second-order in time, and the S 4, S 4c and S 4RK methods are all fourth-order in time cf. Table 5.2 and Figure 5.1 left. ii For any fixed mesh h and time τ, the computational time for S 1 and S 2 are quite similar, the computational time of S 4c, S 4 and S 4RK are about two times, three times and six times of the S 2 method, respectively cf. Table 5.2. iii Among the three fourth-order time-splitting methods, S 4c and S 4RK are quite similar in terms of numerical errors for any fixed τ and they are much smaller than that of the S 4 method, especially when the τ is not so small cf. Table j=0

23 A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation 23 τ 0 = 1/2 τ 0 /2 τ 0 /2 2 τ 0 /2 3 τ 0 /2 4 τ 0 /2 5 τ 0 /2 6 S 1 rate e Φ t = E E E E E E-2 CPU Time S 2 rate e Φ t = E E E E E E E-4 CPU Time S 4 rate e Φ t = E E E E E E E-8 CPU Time S 4c rate e Φ t = E E E E E E E-10 CPU Time S 4RK rate e Φ t = E E E E E E E-10 CPU Time Table 5.2 Temporal errors e Φ t = 6 of different time-splitting methods under different time step τ for the Dirac equation 1.17 in 1D. Here we also list convergence rates and computational time CPU time in seconds for comparison Φ n Φtn, l τ S 1 S 2 S 4 S 4c S 4RK Φ n Φtn, l 2/τ α S 1, α = 1 S 2, α = 2 S 4, α = 4 S 4c, α = 4 S 4RK, α = τ Fig. 5.1 Temporal errors e Φ t = 6 left and e Φ t = 6/τ α with α taken as the order of accuracy of a certain numerical method right of different time-splitting methods under different time step τ for the Dirac equation 1.17 in 1D. 5.2 and Figure 5.1 left. iv For the constants in front of the convergence rates of different methods, C 4 C 1 C 2 Ĉ4 C 4 cf. Figure 5.1 right. v For the S 4 method, it suffers

24 24 Weizhu Bao, Jia Yin Φ n Φtn, l S 1 S 2 S 4 S 4C S 4RK t Fig. 5.2 Time evolution of the errors e Φ t under h= 16 1 and τ = 0.1 over long time of different time-splitting methods for the Dirac equation 1.17 in 1D. from convergence rate reduction when the time step is not small and a very large constant in front of the convergence rate. Thus this method is, in general, to be avoided in practical computation, which has been observed when it is applied for the nonlinear Schrödinger equation too [59]. To compare the long time behavior of different time-splitting methods, Figure 5.2 depicts e Φ t under mesh size h= 1 16 and time step τ = 0.1 for 0 t T := 50. From Figure 5.2, we can observe: i The errors increase very fast when t is small, e.g. 0 t O1, and they almost don t change when t 1, thus they are suitable for long time simulation, especially the fourth-order methods. ii When t is not large, the error of the S 4 method is about 10 times bigger than that of the S 4c method; however, when t 1, it becomes about 100 times larger. iii The error of the S 4RK method is always the smallest among all the time-splitting methods. Based on the efficiency and accuracy as well as long time behavior, in conclusion, for the three fourth-order time-splitting methods, S 4c is more accurate than S 4 and it is more efficient than S 4RK. Thus the S 4c method is highly recommended for studying the dynamics of the Dirac equation, especially in 1D. 5.2 An example in 2D For simplicity, here we only compare the three fourth-order integrators, i.e., S 4c, S 4 and S 4RK via an example in 2D. In order to do so, in the Dirac equation 1.17, we take d = 2,

25 A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation 25 h 0 = 1/2 h 0 /2 h 0 /2 2 h 0 /2 3 S E E E-10 S 4c E E E-10 S 4RK E E E-10 Table 5.3 Spatial errors e Φ t = 2 of different time-splitting methods under different mesh size h for the Dirac equation 1.17 in 2D. = δ = ν = 1 and take the potential in honey-comb form 4π 4π 4π Vx=cos e 1 x + cos e 2 x + cos e 3 x, A 1 x=a 2 x=0, x R 2, 5.4 with e 1 = 1,0 T, e 2 =1/2, 3/2 T, e 3 =1/2, 3/2 T. 5.5 The initial data in 1.18 is taken as: φ 1 0,x=e x2 +y 2 2, φ 2 0,x=e x 12 +y 2 2, x=x,y T R The problem is solved numerically on a bounded domain Ω = 10, 10 10, 10. Similar to the 1D example, we obtain a numerical exact solution by using the S 4c method with a fine mesh size h e = 1 32 and a small time step τ e = The error for the numerical solution Φ n with mesh size h and time step τ is quantified as e Φ t n = Φ n Φt n, l 2 = h M 1 j=0 M 1 Φt n,x j,y l Φ n jl l=0 Similar to the 1D case, in order to compare the spatial errors, we take time step τ = τ e = 10 4 such that the temporal discretization error could be negligible. Table 5.3 lists numerical errors e Φ t = 2 for different time-splitting methods under different mesh size h. In order to compare the temporal errors, we take mesh size h = h e = 32 1 such that the spatial discretization error could be negligible. Table 5.4 lists numerical errors e Φ t = 2 for different time-splitting methods under different time step τ. From Tables 5.3&5.4, we can draw the following conclusions: i All the three methods are spectrally accurate in space and fourth-order in time. ii For any fixed mesh size h and time step τ, the computational times of the S 4 and S 4RK methods are approximately 1.5 times and 3 times more than that of the S 4c method, respectively. iii S 4c and S 4RK are quite similar in terms of numerical errors for any fixed τ and the errors are much smaller than that of the S 4 method, especially when τ is not so small. iv Again, order reduction in time was

26 26 Weizhu Bao, Jia Yin τ 0 = 1/2 τ 0 /2 τ 0 /2 2 τ 0 /2 3 τ 0 /2 4 τ 0 /2 5 τ 0 /2 6 S 4 Order Error 4.33E E E E E E E-8 CPU Time S 4c Order Error 6.75E E E E E E E-9 CPU Time S 4RK Order Error 8.32E E E E E E E-10 CPU Time Table 5.4 Temporal errors e Φ t = 2 of different fourth order time-splitting methods under different time step τ for the Dirac equation 1.17 in 2D. Here we also list convergence rates and computational time CPU time in seconds for comparison. observed in the S 4 method when τ is not small, however, there is almost no order reduction in time for the S 4c and S 4RK methods. Again, based on the efficiency and accuracy for the Dirac equation in high dimensions, in conclusion, for the three fourth-order time-splitting methods, S 4c is more accurate than S 4 and it is more efficient than S 4RK. Thus the S 4c method is highly recommended for studying the dynamics of the Dirac equation in high dimensions, especially without magnetic potential. Based on our numerical observation, we observe numerically that the time-splitting methods for the Dirac equation 1.7 or 1.17 in the nonrelativistic limit regime without magnetic potentials converge uniformly in time with respect to the parameter 0, 1], i.e. they show super-resolution in the sense that time step size τ can be taken much larger than the wavelength in time at O 2. The proof of super-resolution for S 1 and S 2 will be given in a on-going paper [11]. 6 Spatial/temporal resolution of the S 4c method in different parameter regimes In this section, we study numerically temproal/spatial resolution of the fourth-order compact time-splitting Fourier pseudospectral S 4c method 4.11 for the Dirac equation in different parameter regimes. We take d = 1 and the electromagnetic potentials as 5.1 in Dirac equation To quantify the numerical error, we adapt the relative errors of the wave function