Journal of Computational Physics 199 (2004) 663–687 www.elsevier.com/locate/jcp
An efficient and stable numerical method for the Maxwell–Dirac system Weizhu Bao *, Xiang-Gui Li Department of Computational Science, National University of Singapore, Singapore 117543, Singapore Received 14 January 2004; received in revised form 27 February 2004; accepted 9 March 2004 Available online 12 April 2004
Abstract In this paper, we present an explicit, unconditionally stable and accurate numerical method for the Maxwell–Dirac system (MD) and use it to study dynamics of MD. As preparatory steps, we take the three-dimensional (3D) Maxwell– Dirac system, scale it to obtain a two-parameter model and review plane wave solution of free MD. Then we present a time-splitting spectral method (TSSP) for MD. The key point in the numerical method is based on a time-splitting discretization of the Dirac system, and to discretize nonlinear wave-type equations by pseudospectral method for spatial derivatives, and then solving the ordinary differential equations (ODEs) in phase space analytically under appropriate chosen transmission conditions between different time intervals. The method is explicit, unconditionally stable, time reversible, time transverse invariant, and of spectral-order accuracy in space and second-order accuracy in time. Moreover, it conserves the particle density exactly in discretized level and gives exact results for plane wave solution of free MD. Extensive numerical tests are presented to confirm the above properties of the numerical method. Furthermore, the tests also suggest the following meshing strategy (or e-resolution) is admissible in the ‘nonrelativistic’ limit regime (0 < e 1): spatial mesh size h ¼ OðeÞ and time step 4t ¼ Oðe2 Þ, where the parameter e is inversely proportional to the speed of light. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Maxwell–Dirac system; Time-splitting spectral method; Unconditionally stable; Time reversible; Semiclassical; Plane wave
1. Introduction One of the fundamental quantum-relativistic equations is given by the Maxwell–Dirac system (MD), i.e. the Dirac equation [16,28] for the electron as a spinor coupled to the Maxwell equations for the electromagnetic field. It represents the time-evolution of fast (relativistic) electrons and positrons within selfconsistent generated electromagnetic fields. In its most compact form, the Dirac equation reads [8,17,23,27]
*
Corresponding author. Tel.: +65-6874-3337; fax: +65-6774-6756. E-mail addresses:
[email protected] (W. Bao),
[email protected] (X.-G. Li). URL: http://www.cz3.nus.edu.sg/~bao/.
0021-9991/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2004.03.003
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i hcg og m0 c þ ecg Ag W ¼ 0:
ð1:1Þ
Here the unknown W is the 4-vector complex wave function of the ‘‘spinorfield’’: Wðt; xÞ ¼ ðW1 ; W2 ; W3 ; W4 ÞT 2 C4 , x0 ¼ ct, x ¼ ðx1 ; x2 ; x3 ÞT 2 R3 with x0 ; x denoting the time – resp. spatial coordinates in Minkowski space. og stands for oxog , i.e. o0 ¼ oxo0 ¼ 1coto , ok ¼ oxok ðk ¼ 1; 2; 3Þ, where we consequently adopt notation that Greek letter P g denotes 0, 1, 2, 3 and k denotes the three spatial dimension indices 1, 2, 3. cg Ag 3 stands for the summation g¼0 cg Ag . The physical constants are: h for the Planck constant, c for the speed of light, m0 for the electron’s rest mass, and e for the unit charge. By cg 2 C44 , g ¼ 0; . . . ; 3, we denote the 4 4 Dirac matrices given by 0 I2 0 rk c0 ¼ ; ck ¼ ; k ¼ 1; 2; 3; 0 I2 rk 0 where Im (m a positive integer) is the m m identity matrix and rk ðk ¼ 1; 2; 3Þ the 2 2 Pauli matrices, i.e. r1 :¼
0 1 ; 1 0
r2 :¼
0 i
i ; 0
r3 :¼
1 0
0 : 1
Ag ðt; xÞ 2 R, g ¼ 0; . . . ; 3, are the components of the time-dependent electromagnetic potential, in particT ular V ðt; xÞ ¼ A0 ðt; xÞ is the electric potential and Aðt; xÞ ¼ ðA1 ; A2 ; A3 Þ is the magnetic potential vector. Hence the electric and magnetic fields are given by Eðt; xÞ ¼ rA0 ot A ¼ rV ot A;
Bðt; xÞ ¼ curl A ¼ r A:
ð1:2Þ
In order to determine the electric and magnetic potentials from fields uniquely, we have to choose a gauge. In practice, the Lorentz gauge condition is often introduced 1 1 Lðt; xÞ :¼ ot V þ r A ¼ ot A0 þ r A ¼ 0: c c
ð1:3Þ
Thus the electric and magnetic fields are governed by the Maxwell equation: 1 1 J; r B ¼ 0; ot E þ r B ¼ c c0 1 1 ot B þ r E ¼ 0; r E ¼ q; c 0
ð1:4Þ ð1:5Þ T
where 0 is the permittivity of the free space. The particle density q and current density J ¼ ðj1 ; j2 ; j3 Þ are defined as follows: q ¼ ejWj2 :¼ e
4 X
jWj j2 ;
T ak W; jk ¼ echW; ak Wi :¼ ecW
k ¼ 1; 2; 3;
ð1:6Þ
j¼1
where f denotes the conjugate of f and 0 rk ak ¼ c 0 c k ¼ ; k ¼ 1; 2; 3: rk 0
ð1:7Þ
From now on, we adopt the standard notations j j, h; i and k k for l2 -norm of a vector, inner product and L2 -norm of a function.
W. Bao, X.-G. Li / Journal of Computational Physics 199 (2004) 663–687
665
Separating the time derivative associated to the ‘‘relativistic time variable’’ x0 ¼ ct and applying c0 from left of (1.1), plugging (1.2) into (1.4) and (1.5), noticing (1.3), we have the following Maxwell–Dirac system [26] ihot W ¼
3 X
ak ð i hcok eAk ÞW þ eV W þ m0 c2 bW;
ð1:8Þ
k¼1
1 2 1 o D V ¼ q; c2 t 0
1 2 1 o J: D A¼ c2 t c0
ð1:9Þ
The vector wave function W is normalized as kWðt; Þk2 :¼
Z
jWðt; xÞj2 dx ¼ 1:
ð1:10Þ
R3
The MD system (1.8) and (1.9) represents the time-evolution of fast (relativistic) electrons and positrons within self-consistent generated electromagnetic fields. From the mathematical point of view, the strongly nonlinear MD system poses a hard problem in the study of PDEs arising from quantum physics. Well posedness and existence of solutions on all of R3 but only locally in time has been proved almost 40 years ago [11,12,21]. In particular, there are no global existence results without smallness assumptions on the initial data [19,20]. Thus the MD system is quite involved from the numerical point of view as it poses major open problems from analytical point of view. For solitary solution of MD, we refer [1,10,13,14,23]. The aim of this paper is to design an explicit, unconditionally stable and accurate numerical method for the MD system and apply it to study dynamics of MD. The key point in the numerical method is based on a time-splitting discretization of the Dirac system (1.8), which was used successfully to solve nonlinear Schr€ odinger equation (NLS) [2–5] and Zakharov system [6,7], and to discretize the nonlinear wave-type equation (1.9) by pseudospectral method for spatial derivatives, and then solving the ODEs in phase space analytically under appropriate chosen transmission conditions between different time intervals. The paper is organized as follows. In Section 2, we start out with the MD, scale it to get a two-parameter model and review plane wave solution of free MD. In Section 3, we present a time-splitting spectral method (TSSP) for the MD and show some properties of the numerical method. In Section 4, numerical tests of MD for different cases are reported to demonstrate efficiency and high resolution of our numerical method. In Section 5 a summary is given.
2. The Maxwell–Dirac system Consider the Maxwell–Dirac system represents the time-evolution of fast (relativistic) electrons and positrons within external and self-consistent generated electromagnetic fields [26] ihot W ¼
3 X
hcok e Ak þ Aext W þ eðV þ V ext ÞW þ m0 c2 bW; ak i k
ð2:1Þ
k¼1
1 2 1 o D V ¼ q; c2 t 0
1 2 1 o D A¼ J; c2 t c0
ð2:2Þ
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W. Bao, X.-G. Li / Journal of Computational Physics 199 (2004) 663–687 T
3 ext ext where V ext ¼ V ext ðt; xÞ 2 R and Aext ðt; xÞ ¼ ðAext 1 ; A2 ; A3 Þ 2 R are the external electric and magnetic potentials, respectively.
2.1. Dimensionless Maxwell–Dirac system We rescale the MD (2.1) and (2.2) under the normalization (1.10) by introducing a reference velocity v, length L ¼ e2 =m0 v2 0 , time T ¼ v=L, and strength of the electromagnetic potential k ¼ e=L0 , as x ~¼ ; x L
~t ¼
t ; T
~ ~t; x ~Þ ¼ L3=2 Wðt; xÞ; Wð
~ ~t; x ~Þ ¼ kAðt; xÞ; Að
~Þ ¼ kV ðt; xÞ; V~ ð~t; x
~ ext ð~t; x ~Þ ¼ kAext ðt; xÞ; A
~Þ ¼ kV ext ðt; xÞ: V~ ext ð~t; x
ð2:3Þ ð2:4Þ
Plugging (2.3) and (2.4) into (2.1) and (2.2), then removing all , we get the following dimensionless MD: idot W ¼ i
3 3 X dX 1 ext ak ok W ak ðAk þ Aext ÞW þ 2 bW; k ÞW þ ðV þ V e k¼1 e k¼1
e2 o2t D V ¼ q;
e2 o2t D A ¼ eJ:
ð2:5Þ
ð2:6Þ
Two important dimensionless parameters in the MD (2.5) and (2.6) are given by the ratio of the reference velocity to the speed of light, i.e. e, and the scaled Planck constant, i.e. d, as v e :¼ ; c
d :¼
0 v h : e2
ð2:7Þ
The position and current densities, Lorentz gauge, as well as electric and magnetic fields in dimensionless variables are 1 jk ðt; xÞ ¼ hWðt; xÞ; ak Wðt; xÞi; k ¼ 1; 2; 3; e Lðt; xÞ ¼ eot V ðt; xÞ þ r Aðt; xÞ; t P 0; x 2 R3 ; Eðt; xÞ ¼ eot Aðt; xÞ rV ðt; xÞ; Bðt; xÞ ¼ r Aðt; xÞ:
qðt; xÞ ¼ jWðt; xÞj2 ;
ð2:8Þ ð2:9Þ ð2:10Þ
When v c and choosing v ¼ c, then e ¼ 1 in (2.7) and the MD (2.5) and (2.6) collapse to a one-parameter model which is used in [26] to study classical limit and semiclassical asymptotics of MD. In this case, the parameter d is the same as the canonical parameter a used in physical literatures [16,28]. When v c and choosing v ¼ e2 = h0 , then d ¼ 1 and 0 < e 1 in (2.7), again the MD (2.5) and (2.6) collapse to a oneparameter model which is called as ‘nonrelativistic’ limit regime and used in [8,9,18,22,24,25] to study seminonrelativistic limits of MD, i.e. letting e ! 0 in (2.5) and (2.6). For electrons, e ¼ 1 and d 10:9149 [26]. The MD system (2.5) and (2.6) together with initial data Z 2 ð0Þ ð0Þ Wð0; xÞ ¼ W ðxÞ with kW k ¼ jWð0Þ ðxÞj dx ¼ 1; ð2:11Þ R3
V ð0; xÞ ¼ V ð0Þ ðxÞ;
ot V ð0; xÞ ¼ V ð1Þ ðxÞ;
Að0; xÞ ¼ Að0Þ ðxÞ;
ot Að0; xÞ ¼ Að1Þ ðxÞ
x 2 R3 ;
ð2:12Þ ð2:13Þ
W. Bao, X.-G. Li / Journal of Computational Physics 199 (2004) 663–687
667
is time-reversible and time-transverse invariant, i.e., if constants a0 and a1 are added to V ð0Þ and V ð1Þ , respectively, in (2.12), then the solution V get added by a0 þ a1 t and W get multiplied by eitða0 þa1 t=2Þ=d , which and taking leaves density of each particle jwj j (j ¼ 1; 2; 3; 4) unchanged. Moreover, multiplying (2.5) by W imaginary parts we obtain the conservation law ot qðt; xÞ þ r Jðt; xÞ ¼ 0;
t P 0; x 2 R3 :
ð2:14Þ
From (2.14) and (2.6), we get the Lorentz gauge of the MD system (2.5) and (2.6) satisfying 2 2 e ot D Lðt; xÞ ¼ eðot q þ r JÞ ¼ 0; t P 0; x 2 R3 ; Lð0; xÞ ¼ eot V ð0; xÞ þ r Að0; xÞ ¼ eV ð1Þ ðxÞ þ r Að0Þ ðxÞ;
ð2:15Þ ð2:16Þ
1 ot Lð0; xÞ ¼ eott V ð0; xÞ þ r ot Að0; xÞ ¼ ½qð0; xÞ þ DV ð0; xÞ þ r Að1Þ ðxÞ e i 1 h ð0Þ 2 ¼ DV ðxÞ þ jWð0Þ ðxÞj þ er Að1Þ ðxÞ ; x 2 R3 : e
ð2:17Þ
Thus if the initial data in (2.11)–(2.13) satisfy eV ð1Þ ðxÞ þ r Að0Þ ðxÞ 0;
2
DV ð0Þ ðxÞ þ jWð0Þ ðxÞj þ er Að1Þ ðxÞ 0;
x 2 R3 ;
ð2:18Þ
which implies Lð0; xÞ ¼ 0;
ot Lð0; xÞ ¼ 0;
x 2 R3 ;
ð2:19Þ
the gauge is henceforth conserved during the time-evolution of the MD (2.5) and (2.6). 2.2. Plane wave solution If the initial data in (2.11)–(2.13) for the MD (2.5) and (2.6) are chosen as Wð0Þ ðxÞ ¼ Wð0Þ eixx ¼ Wð0Þ eiðx1 x1 þx2 x2 þx3 x3 Þ ;
ð2:20Þ
V ð0Þ ðxÞ V ð0Þ ;
ð2:21Þ
V ð1Þ ðxÞ V ð1Þ ; 0
Að0Þ ðxÞ Að0Þ
1 ð0Þ A1 B C ; ¼ @ Að0Þ 2 A
x 2 R3 ;
ð0Þ A3
0
Að1Þ ðxÞ Að1Þ
1 ð1Þ A1 B C ; ¼ @ Að1Þ 2 A
ð2:22Þ
ð1Þ A3
where x ¼ ðx1 ; x2 ; x3 ÞT with xj (j ¼ 1; 2; 3) integers, V ð0Þ , V ð1Þ constants, Wð0Þ , Að0Þ , Að1Þ constant vectors, and 0
Wð0Þ
1 x3 B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C x1 þ ixffi 2 1 B C rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ d2 þ jxj2 d1 A; pffiffiffi 2 2 2 1 2 4p p d þ jxj d d þ jxj 0
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W. Bao, X.-G. Li / Journal of Computational Physics 199 (2004) 663–687
then the MD (2.5) and (2.6) with e ¼ 1, Aext ¼ A and V ext ¼ V , i.e. free MD [15], admits the following plane wave solution: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð0Þ Wðt; xÞ ¼ W exp ix x it d2 þ jxj ; ð2:23Þ
V ðt; xÞ ¼ V ext ¼ V ð0Þ þ V ð1Þ t þ
1 2 t ; 16p3
x 2 R3 ; t P 0;
ð2:24Þ
1 Aðt; xÞ ¼ Aext ¼ Að0Þ þ Að1Þ t þ Jð0Þ t2 ; 2 ð0Þ
ð0Þ
ð2:25Þ
ð0Þ T
where Jð0Þ ¼ ðj1 ; j2 ; j3 Þ and ð0Þ
jk ¼
xk qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 8p3 d2 þ jxj
k ¼ 1; 2; 3:
Here the normalization condition for the wave function is set as Z p Z p Z p 2 jWðt; xÞj dx ¼ 1: p
p
p
3. Numerical method In this section we present an explicit, unconditionally stable and accurate numerical method for the MD (2.5) and (2.6). We shall introduce the method in 3D on a box with periodic boundary conditions. For 3D in a box X ¼ ½a1 ; b1 ½a2 ; b2 ½a3 ; b3 , the problem with initial and boundary conditions become idot W ¼ i
3 3 X dX 1 ext ak ok W ak ðAk þ Aext ÞW þ 2 bW; k ÞW þ ðV þ V e k¼1 e k¼1
e2 o2t D V ðt; xÞ ¼ q;
e2 o2t D Aðt; xÞ ¼ eJ;
Wð0; xÞ ¼ Wð0Þ ðxÞ;
V ð0; xÞ ¼ V ð0Þ ðxÞ;
Að0; xÞ ¼ Að0Þ ðxÞ;
ot Að0; xÞ ¼ Að1Þ ðxÞ;
x 2 X;
ot V ð0; xÞ ¼ V ð1Þ ðxÞ; x2X
with periodic boundary conditions for W; V ; A on oX; where V ð1Þ and A
ð0Þ
t > 0;
ð3:1Þ
ð3:2Þ ð3:3Þ ð3:4Þ ð3:5Þ
satisfy
eV ð1Þ ðxÞ þ r Að0Þ ðxÞ ¼ 0;
x2X
and the normalization condition for the wave function is set as Z Z 2 2 2 jWðt; xÞj dx ¼ jWð0Þ ðxÞj dx ¼ 1: kWðt; Þk :¼ X
X
ð3:6Þ
ð3:7Þ
W. Bao, X.-G. Li / Journal of Computational Physics 199 (2004) 663–687
669
Moreover, integrating the first equation in (3.2) we obtain e2
d2 dt2
Z
V ðt; xÞ dx ¼
X
Z
qðt; xÞ dx ¼
Z
X
2
jWðt; xÞj dx ¼ 1;
t P 0:
ð3:8Þ
X
This implies that MeanðV ðt; ÞÞ ¼ MeanðV ð0Þ Þ þ t MeanðV ð1Þ Þ þ
t2 ; 2e2
t P 0;
ð3:9Þ
where Meanðf Þ :¼
Z
f ðxÞ dx: X
3.1. Time-splitting spectral discretization We choose the spatial mesh size hj ¼ Dt. Denote the grid points as xp;q;r ¼ ðx1;p ; x2;q ; x3;r Þ
T
bj aj Mj
(j ¼ 1; 2; 3) in xj -direction with Mj given integer and time step
ðp; q; rÞ 2 N;
where N ¼ fðp; q; rÞ j 0 6 p 6 M1 ; 0 6 q 6 M2 ; 0 6 r 6 M3 g; x1;p ¼ a1 þ ph1 ; x2;q ¼ a2 þ qh2 ; x3;r ¼ a3 þ rh3 ðp; q; rÞ 2 N and time step as tn ¼ nDt;
tnþ1=2 ¼ ðn þ 1=2ÞDt;
n ¼ 0; 1; . . .
n Let Wnp;q;r , Vp;q;r and Anp;q;r be the numerical approximation of Wðtn ; xp;q;r Þ, V ðtn ; xp;q;r Þ and Aðtn ; xp;q;r Þ, ren spectively. Furthermore, let Wn , V n and An be the solution vector at time t ¼ tn with components Wnp;q;r , Vp;q;r n and Ap;q;r , respectively. From time t ¼ tn to t ¼ tnþ1 , we discretize the MD (3.1) and (3.2) as follows: The nonlinear wave-type equations (3.2) are discretized by pseudospectral method for spatial derivatives and then solving the ODEs in phase space analytically under appropriate chosen transmission conditions between different time intervals, and the Dirac equation (3.1) is solved in two splitting steps. For the nonlinear wave-type equations (3.2), we assume
V ðt; xÞ ¼
X
n Vej;k;l ðtÞeilj;k;l ðxaÞ ;
x 2 X; tn 6 t 6 tnþ1 ;
ðj;k;lÞ2M
where fe denotes the Fourier coefficients of f and M¼
M1 M1 M2 M2 M3 M3 6j< ; 6k < ; 6l < ðj; k; lÞ j ; 2 2 2 2 2 2
ð3:10Þ
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W. Bao, X.-G. Li / Journal of Computational Physics 199 (2004) 663–687
0
lj;k;l
1 ð1Þ lj B C ¼ @ lð2Þ A; k ð3Þ ll
0
1 a1 a ¼ @ a2 A a3
with ð1Þ
lj ¼
2pj ; b1 a1
ð2Þ
lk ¼
2pk ; b2 a2
ð3Þ
ll ¼
2pl ; b3 a3
ðj; k; lÞ 2 M:
Plugging (3.10) and (2.8) into (3.2), noticing the orthogonality of the Fourier series, we get the following ODEs for n P 0: e2
n d2 Vej;k;l ðtÞ g2 2 n þ jlj;k;l j Vej;k;l ðtÞ ¼ e q j;k;l ðtn Þ :¼ ðjWn j Þ j;k;l ; 2 dt
( n Vej;k;l ðtn Þ
¼
ð0Þ Þ ðVg j;k;l ; n1 Vej;k;l ðtn Þ;
n ¼ 0; n > 0;
tn 6 t 6 tnþ1 ;
ðj; k; lÞ 2 M:
ð3:11Þ
ð3:12Þ
As noticed in [6], for each fixed ðj; k; lÞ 2 M, Eq. (3.11) is a second-order ODE. It needs two initial conditions such that the solution is unique. When n ¼ 0 in (3.11) and (3.12), we have the initial condition (3.12) and we can pose the other initial condition for (3.11) due to the initial condition (3.3) for the MD (3.1) and (3.2) d e0 d 0 ð1Þ Þ V j;k;l ðt0 Þ ¼ Vej;k;l ð0Þ ¼ ðVg j;k;l : dt dt
ð3:13Þ
Then the solution of (3.11), (3.12) and (3.13) for t 2 ½0; t1 is
0 Vej;k;l ðtÞ ¼
8 g ð0Þ 2 2 2 g g > ð0Þ ð1Þ > > ðV Þ j;k;l þ t ðV Þ j;k;l þ ðjW j Þ j;k;l t =2e ; > > > > g2 2 ð0Þ Þ < ðVg ðjWð0Þ j Þ =jlj;k;l j cosðtjlj;k;l j=eÞ j;k;l
> > > > > > > :
j ¼ k ¼ l ¼ 0;
j;k;l
e ð1Þ Þ þ ðVg j;k;l sinðtjlj;k;l j=eÞ jlj;k;l j g þ ðjWð0Þ j2 Þ j;k;l =jlj;k;l j2
otherwise:
But when n > 0, we only have one initial condition (3.12). One cannot simply pose the continuity between n1 and dtd Vej;k;l ðtÞ across the time t ¼ tn due to the right-hand side in (3.11) is usually different in two g2 g2 adjacent time intervals ½tn1 ; tn and ½tn ; tnþ1 , i.e. e q j;k;l ðtn1 Þ ¼ ðjWn1 j Þ j;k;l 6¼ ðjWn j Þ j;k;l ¼ e q j;k;l ðtn Þ. Since our goal is to develop explicit scheme and we need linearize the nonlinear term in (3.2) in our discretization (3.11), in general,
d en V ðtÞ dt j;k;l
d e n1 d n1 d n d n þ V ðt Þ ¼ lim Vej;k;l ðtÞ 6¼ limþ Vej;k;l ðtÞ ¼ Vej;k;l ðtm Þ; t!tn dt dt j;k;l n dt t!tn dt
n ¼ 1; . . . ðj; k; lÞ 2 M:
ð3:14Þ
n n1 ðtnþ Þ dtd Vej;k;l ðtn Þ across the time t ¼ tn . In order to get a Unfortunately, we do not know the jump dtd Vej;k;l unique solution of (3.11) and (3.12) for n > 0, here we pose an additional condition:
W. Bao, X.-G. Li / Journal of Computational Physics 199 (2004) 663–687 n n1 Vej;k;l ðtn1 Þ ¼ Vej;k;l ðtn1 Þ ðj; k; lÞ 2 M:
671
ð3:15Þ
n The condition (3.15) is equivalent to pose the solution Vej;k;l ðtÞ on the time interval ½tn ; tnþ1 of (3.11) and (3.12) is also continuity at the time t ¼ tn1 . After a simple computation, we get the solution of (3.11), (3.12) and (3.15) for n > 0
8 n 2 q j;k;l ðtn Þðt tn Þ =2e2 Vej;k;l ðtn Þ þ e > > h i > > 2 ttn e n > 2 e n1 > V þ ðt q ðt n Þ V j;k;l ðtn1 Þ þ e n ÞðDtÞ =2e ; > j;k;l j;k;l Dt >h i > > > 2 < Ve n e q ðt Þ=jl j cosððt tn Þjlj;k;l j=eÞ n j;k;l j;k;l j;k;l n h Vej;k;l ðtÞ ¼ n1 > > q j;k;l ðtn Þ=jlj;k;l j2 Vej;k;l ðtn1 Þ þ ð1 cosðjlj;k;l jDt=eÞÞe > > i > > sinððttn Þjlj;k;l j=eÞ > n > þ Vej;k;l ðtn Þ cosðjlj;k;l jDt=eÞ sinðjlj;k;l jDt=eÞ > > > : þe q j;k;l ðtn Þ=jlj;k;l j2
j ¼ k ¼ l ¼ 0;
otherwise:
Discretization for the equation of A in (3.2) can be done in a similar way. For the Dirac equation (3.1), we solve it in two splitting steps. One solves first idot Wðt; xÞ ¼ i
3 dX 1 ak ok W þ 2 bW; e k¼1 e
x 2 X; tn 6 t 6 tnþ1
ð3:16Þ
for the time step of length Dt, followed by solving idot Wðt; xÞ ¼ ðV þ V ext ÞW
3 X
ak ðAk þ Aext k ÞW ¼ Gðt; xÞW
ð3:17Þ
k¼1
for the same time step with " Gðt; xÞ ¼ ðV ðt; xÞ þ V
ext
ðt; xÞÞI4
3 X
# ext a Ak ðt; xÞ þ Ak ðt; xÞ : k
ð3:18Þ
k¼1
For each fixed x 2 X, integrating (3.17) from tn to tnþ1 , and then approximating the integral on ½tn ; tnþ1 via the Simpson rule, one reads
Z 1 tnþ1 Wðtnþ1 ; xÞ ¼ exp i Gðt; xÞ dt Wðtn ; xÞ d tn
Dt Gðtn ; xÞ þ 4Gðtnþ1=2 ; xÞ þ Gðtnþ1 ; xÞ =6 Wðtn ; xÞ exp i d
Dt ¼ exp i Gnþ1=2 ðxÞ Wðtn ; xÞ: d
ð3:19Þ
nþ1=2 ðxÞÞT ¼ Gnþ1=2 ðxÞ, it is diagonalizable (see detail in Appendix A), Since Gnþ1=2 ðxÞ is a U-matrix, i.e. ðG i.e. there exist a diagonal matrix Dnþ1=2 ðxÞ and a complex orthogonormal matrix P nþ1=2 ðxÞ, i.e. ðP nþ1=2 ðxÞÞT ¼ ðP nþ1=2 ðxÞÞ1 , such that T
Gnþ1=2 ðxÞ ¼ P nþ1=2 ðxÞDnþ1=2 ðxÞðP nþ1=2 ðxÞÞ ;
x 2 X:
ð3:20Þ
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Plugging (3.20) into (3.19), we obtain
Dt nþ1=2 T nþ1=2 Wðtnþ1 ; xÞ ¼ P ðxÞ exp i D ðxÞ ðP nþ1=2 ðxÞÞ Wðtn ; xÞ; d For discretizing (3.16), we assume X e j;k;l ðtÞeilj;k;l ðxaÞ ; Wðt; xÞ ¼ W
x 2 X:
ð3:21Þ
x 2 X; tn 6 t 6 tnþ1 :
ð3:22Þ
tn 6 t 6 tnþ1 ðj; k; lÞ 2 M;
ð3:23Þ
ðj;k;lÞ2M
Substituting (3.22) into (3.16), we have e j;k;l ðtÞ dW i e j;k;l ðtÞ; ¼ Mj;k;l W dt e where the matrix ð1Þ
ð2Þ
ð3Þ
Mj;k;l ¼ lj a1 þ lk a2 þ ll a3 þ e1 d1 :
ð3:24Þ
Since Mj;k;l is a U-matrix, again it is diagonalizable (see detail in Appendix B), i.e. there exist a diagonal matrix Dj;k;l and a complex orthogonormal matrix Pj;k;l such that T Mj;k;l ¼ Pj;k;l Dj;k;l ðPj;k;l Þ
ðj; k; lÞ 2 M:
ð3:25Þ
Thus the solution of (3.23) is
i e e j;k;l ðtn Þ W j;k;l ðtÞ ¼ exp ðt tn ÞMj;k;l W e
i Te ¼ Pj;k;l exp ðt tn ÞDj;k;l ðPj;k;l Þ W j;k;l ðtn Þ; e
tn 6 t 6 tnþ1 :
From time t ¼ tn to t ¼ tnþ1 , we combine the splitting steps via the standard Strang splitting: X nþ1 n Vej;k;l Vp;q;r ¼ ðtnþ1 Þeilj;k;l ðxp;q;r aÞ ;
ð3:26Þ
ð3:27Þ
ðj;k;lÞ2M
Anþ1 p;q;r ¼
X
e n ðtnþ1 Þeilj;k;l ðxp;q;r aÞ A j;k;l
ðp; q; rÞ 2 N;
ð3:28Þ
ðj;k;lÞ2M
iDt T g n Dj;k;l ðPj;k;l Þ ðW Pj;k;l exp Þ j;k;l eilj;k;l ðxp;q;r aÞ ; 2e ðj;k;lÞ2M
Dt ¼ P nþ1=2 ðxp;q;r Þ exp i Dnþ1=2 ðxp;q;r Þ ðP nþ1=2 ðxp;q;r ÞÞT W p;q;r ; d
X iDt T g Dj;k;l ðPj;k;l Þ ðW ¼ Pj;k;l exp Þ j;k;l eilj;k;l ðxp;q;r aÞ ; 2e ðj;k;lÞ2M
W p;q;r ¼ W p;q;r nþ1 Wp;q;r
X
ð3:29Þ
n e n ðtnþ1 Þ are given in Appendix C and U e j;k;l the discrete Fourier where the formula for Vej;k;l ðtnþ1 Þ and A j;k;l transform coefficients of the vector fUp;q;r ; ðp; q; rÞ 2 Ng are defined as
W. Bao, X.-G. Li / Journal of Computational Physics 199 (2004) 663–687
e j;k;l ¼ U
X 1 Up;q;r eilj;k;l ðxp;q;r aÞ M1 M2 M3 ðp;q;rÞ2Q
ðj; k; lÞ 2 M;
673
ð3:30Þ
where Q ¼ fðp; q; rÞ j 0 6 p 6 M1 1; 0 6 q 6 M2 1; 0 6 r 6 M3 1g: The initial conditions (3.3) and (3.4) are discretized as W0p;q;r ¼ Wð0Þ ðxp;q;r Þ;
0 Vp;q;r ¼ V ð0Þ ðxp;q;r Þ;
dA0p;q;r ð0Þ ¼ Að1Þ ðxp;q;r Þ; dt
0 dVp;q;r ð0Þ ¼ V ð1Þ ðxp;q;r Þ; dt
A0p;q;r ¼ Að0Þ ðxp;q;r Þ;
ðp; q; rÞ 2 N:
Remark 3.1. We use the Simpson quadrature rule to approximate the integration in (3.19) instead of the trapezodial rule which was used in [6,7] for a similar integration. The reason is that we want the quadrature is exact when the MD system (2.5) and (2.6) admits the plane wave solution (2.23)–(2.25). In this case, the integrand Gðt; xÞ is quadratic in t. Thus the algorithm (3.27)–(3.29) gives exact results when the MD system admits plane wave solution. 3.2. Properties of the numerical method 1. Plane wave solution: If the initial data in (3.3) and (3.4) are chosen as in (2.20)–(2.22), and the external electric and magnetic fields, i.e. V ext and Aext , are chosen as in (2.24) and (2.25), then the MD system (3.1)– (3.5) admits the plane wave solution (2.23)–(2.25). It is easy to see that in this case our numerical method (3.27)–(3.29) gives exact results provided that Mj P 2ðjxj j þ 1Þ (j ¼ 1; 2; 3). 2. Time transverse invariant: If constants a0 and a1 are added to V ð0Þ and V ð1Þ , respectively, in (3.3), then the solution V n get added by a0 þ a1 tn and Wn get multiplied by eitn ða0 þa1 tn =2Þ=d , which leaves density of each particle jwnj j (j ¼ 1; 2; 3; 4) unchanged. 3. Conservation: Let U ¼ fUp;q;r ; ðp; q; rÞ 2 Ng and f ðxÞ a periodic function on the box X, and let k kl2 be the usual discrete l2 -norm on the box X, i.e. X kUk2l2 ¼ h1 h2 h3 jUp;q;r j2 ; ð3:31Þ ðp;q;rÞ2Q
DMeanðUÞ ¼ h1 h2 h3
X
Up;q;r ;
ð3:32Þ
ðp;q;rÞ2Q
kf k2l2 ¼ h1 h2 h3
X
jf ðxp;q;r Þj2 :
ð3:33Þ
ðp;q;rÞ2Q
Then we have: Theorem 3.1. The time splitting spectral method (3.27)–(3.29) for the MD conserves the following quantities in the discretized level: kWn kl2 ¼ kW0 kl2 ¼ kWð0Þ kl2 ;
n ¼ 0; 1; 2; . . . ;
ð3:34Þ
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DMeanðV n Þ ¼ DMeanðV ð0Þ Þ þ tn DMeanðV ð1Þ Þ þ Proof. See Appendix D.
tn2 2 DMeanðjWð0Þ j Þ: 2e2
ð3:35Þ
4. Unconditional stability: By the standard Von Neumann analysis for (3.27) and (3.28), noting (3.34), we get the method (3.27)–(3.29) is unconditionally stable. In fact, setting Wn ¼ 0 and plugging n n n1 V~j;k;l ðtnþ1 Þ ¼ lV~j;k;l ðtn Þ ¼ l2 V~j;k;l ðtn1 Þ into (C.3) with jlj the amplification factor, we obtain the characteristic equation: l2 2l cosðjlj;k;l jDt=eÞ þ 1 ¼ 0;
ðj; k; lÞ 2 M:
ð3:36Þ
This implies l ¼ cosðjlj;k;l jDt=eÞ i sinðjlj;k;l jDt=eÞ:
ð3:37Þ
Thus the amplification factor Gj;k;l
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ jlj ¼ cos2 ðjlj;k;l jDt=eÞ þ sin2 ðjlj;k;l jDt=eÞ ¼ 1
ðj; k; lÞ 2 M:
ð3:38Þ
Similar results can be obtained for (3.28). These together with (3.34) imply the method (3.27)–(3.29) is unconditionally stable. This is confirmed by our numerical results in the next section. 5. e-resolution in the ‘nonrelatistic’ limit regime (0 < e 1): As our numerical results in the next section suggest: The meshing strategy (or e-resolution) which guarantees ‘good’ numerical results in the ‘nonrelatistic’ limit regime, i.e. 0 < e 1, is h ¼ maxfh1 ; h2 ; h3 g ¼ OðeÞ;
Dt ¼ Oðe2 Þ:
ð3:39Þ
3.3. For homogeneous Dirichlet boundary conditions In some cases, the periodic boundary conditions (3.5) may be replaced by the following homogeneous Dirichlet boundary conditions: Wðt; xÞ ¼ V ðt; xÞ ¼ 0;
Aðt; xÞ ¼ 0;
x 2 oX; t P 0:
ð3:40Þ
In this case, the method designed above is still valid provided that we replace the Fourier basis functions by sine basis functions. Let M ¼ fðj; k; lÞ j 1 6 j 6 M1 1; 1 6 k 6 M2 1; 1 6 l 6 M3 1g; ð1Þ
lj ¼
pj ; b1 a1
ð2Þ
lk ¼
pk ; b2 a2
ð3Þ
ll ¼
pl b3 a3
ðj; k; lÞ 2 M:
ð3:41Þ
W. Bao, X.-G. Li / Journal of Computational Physics 199 (2004) 663–687
675
The detailed scheme is: pjp qkp rlp ¼ sin sin ; M1 M2 M3 ðj;k;lÞ2M X pjp qkp rlp nþ1 n e sin sin ðp; q; rÞ 2 M; Ap;q;r ¼ A j;k;l ðtnþ1 Þ sin M1 M2 M3 ðj;k;lÞ2M
X iDt pjp qkp rlp T g n Dj;k;l ðPj;k;l Þ ðW Þ j;k;l sin Pj;k;l exp sin sin ; Wp;q;r ¼ 2e M1 M2 M3 ðj;k;lÞ2M
Dt nþ1=2 T nþ1=2 ðxp;q;r Þ exp i D ðxp;q;r Þ ðP nþ1=2 ðxp;q;r ÞÞ W p;q;r ; Wp;q;r ¼ P d
X iDt pjp qkp rlp T g nþ1 Dj;k;l ðPj;k;l Þ ðW Þ j;k;l sin Pj;k;l exp Wp;q;r ¼ sin sin ; 2e M1 M2 M3 ðj;k;lÞ2M
nþ1 Vp;q;r
X
n Vej;k;l ðtnþ1 Þ sin
n e n ðtnþ1 Þ are given in Appendix C with l is replaced by (3.41), and ðtnþ1 Þ and A where the formula for Vej;k;l j;k;l j;k;l e j;k;l the discrete sine transform coefficients of the vector fUp;q;r ; ðp; q; rÞ 2 Ng are defined as U X 8 pjp qkp rlp e U j;k;l ¼ Up;q;r sin sin sin ; ðj; k; lÞ 2 M: ð3:42Þ M1 M2 M3 ðp;q;rÞ2M M1 M2 M3
4. Numerical results In this section, we present numerical results to demonstrate ‘good’ properties of our numerical method for MD and apply it to study dynamics of MD. In Examples 1 and 3, the initial data in (3.3) and (3.4) are chosen as wjð0Þ ðxÞ ¼
ðc1 c2 c3 Þ1=4 exp½ðc1 x21 þ c2 x22 þ c3 x23 Þ=2Þ expðicj x1 =eÞ; 2p3=4
V ð0Þ ðxÞ ¼ 0;
V ð1Þ ðxÞ ¼ 0;
Að0Þ ðxÞ ¼ 0;
x 2 R3 :
ð4:1Þ ð4:2Þ
They, together with Að1Þ , decay to zero sufficient fast as jxj ! 1. This Gaussian-type initial data is often used to study wave motion and interaction in physical literatures. We always compute on a box, which is large enough such that the periodic boundary conditions (3.5) do not introduce a significant aliasing error relative to the problem in the whole space. In our computations, we always choose uniform mesh, i.e. h ¼ h1 ¼ h2 ¼ h3 . 4.1. Numerical accuracy Example 1. Accuracy test and meshing strategy, i.e. we choose d ¼ 1, V ext ðt; xÞ 0, Aext ðt; xÞ 0 in (3.1), c1 ¼ c2 ¼ c3 ¼ 5 and c1 ¼ c2 ¼ c3 ¼ c4 ¼ 1 in (4.1) and Að1Þ ðxÞ ¼ 0 in (3.4). We solve the MD (3.1)–(3.5) on a box X ¼ ½4; 43 by using our numerical method (3.27)–(3.29), and present results for two different regimes of velocity, i.e. 1=e: Case I. Oð1Þ-velocity speed, i.e. we choose e ¼ 1 in (3.1), (3.2) and (4.1). Here we test the spatial and temporal discretization errors. Let W, V and A be the ‘exact’ solutions which are obtained numerically by
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W. Bao, X.-G. Li / Journal of Computational Physics 199 (2004) 663–687
using our numerical method with a very fine mesh and time step, e.g. h ¼ 18 and Dt ¼ 0:0001, and Wh;Dt , V h;Dt and Ah;Dt be the numerical solution obtained by using our method with mesh size h and time step Dt. To quantify the numerical method, we define the error functions as eW ðtÞ ¼ kWðt; Þ Wh;Dt ðt; Þkl2 ;
eA ðtÞ ¼ kAðt; Þ Ah;Dt ðt; Þkl2 ;
eV ðtÞ ¼ kV ðt; Þ V h;Dt ðt; Þkl2 :
First, we test the discretization error in space. In order to do this, we choose a very small time step, e.g. Dt ¼ 0:0001, such that the error from time discretization is negligible comparing to the spatial discretization error. Table 1 lists the numerical errors of eW ðtÞ, eV ðtÞ and eA ðtÞ at t ¼ 0:4 with different mesh sizes h. Second, we test the discretization error in time. Table 2 shows the numerical errors of eW ðtÞ, eV ðtÞ and eA ðtÞ at t ¼ 0:4 under different time step Dt and mesh size h ¼ 1=4. Third, we test the density conservation in (3.34). Table 3 shows kWkl2 at different times. Case II: ‘nonrelativistic’ limit regime, i.e. 0 < e 1. Here we test the e-resolution of our numerical method. Fig. 1 shows the numerical results at t ¼ 0:4 when we choose the meshing strategy: e ¼ 1, h ¼ 1=2, Dt ¼ 0:2; e ¼ 1=2, h ¼ 1=4, Dt ¼ 0:05; e ¼ 1=4, h ¼ 1=8, Dt ¼ 0:0125; which corresponds to meshing strategy h ¼ OðeÞ, Dt ¼ Oðe2 Þ. From Tables 1–3 and Fig. 1, we can draw the following observations: Our numerical method for MD is of spectral order accuracy in space and second order accuracy in time, and conserves the density up to 12-digits. In the ‘nonrelativistic’ limit regime, i.e. 0 < e 1, the e-resolution is: h ¼ OðeÞ and Dt ¼ Oðe2 Þ. Furthermore, our additional numerical experiments confirm that the method is unconditionally stable, and show that meshing strategy: h ¼ OðeÞ and Dt ¼ OðeÞ gives ‘incorrect’ numerical results in ‘nonrelativistic’ limit regime.
Table 1 Spatial discretization error analysis: at time t ¼ 0:4 under Dt ¼ 0:0001 Mesh
h¼1
h ¼ 0:5
h ¼ 0:25
h ¼ 0:125
eW ðtÞ eV ðtÞ eA ðtÞ
0:76250 6:4937E 3 7:0499E 3
5:8928E 2 3:9210E 4 4:6114E 4
7:7029E 6 8:8440E 5 1:1609E 6
2:2164E 11 7:6842E 13 7:6508E 13
Table 2 Temporal discretization error analysis: at time t ¼ 0:4 under h ¼ 1=4 Time step
Dt ¼ 0:05
Dt ¼ 0:025
Dt ¼ 0:0125
Dt ¼ 0:00625
eW ðtÞ eV ðtÞ eA ðtÞ
7:7643E 5 2:9906E 4 3:5809E 4
1:9592E 5 7:4114E 5 8:8633E 5
4:9041E 6 1:8405E 5 2:2004E 5
1:2063E 6 4:5103E 6 5:381E 6
Table 3 Density conservation test Time
t¼0
t¼1
t¼2
kWðt; Þkl2
0.9999999999999
0.9999999999998
0.9999999999996
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677
0.25 0.09 0.08
0.2
1 1
A (x ,0,0)
|ψ1(x1,0,0)|
2
0.07 0.15
0.1
0.06 0.05 0.04 0.03 0.02
0.05
0.01 0
0 –4
–2
(a)
0
x
2
–0.01 –4
4
1
0.25
1 1
A (x ,0,0)
1 1
|ψ (x ,0,0)| 2
4
2
4
2
4
0.02 0.15
0.1
–4
0.015 0.01 0.005
0
0 –2
0
2
4
x1
(b)
–4
0.025
0.2
0.02
0.15
0.015
1 1
0.1
0.01
0.05
0.005
0
0 –2
0
x
1
2
4
–4
(f)
0
x1
0.25
–4
–2
(e)
A (x ,0,0)
2
2
1
0.025
0.05
|ψ1(x1,0,0)|
0
x 0.03
0.2
(c)
–2
(d)
–2
0
x
1
Fig. 1. Meshing strategy test in Example 1 for wave function jW1 ðt; x1 ; 0; 0Þj2 (left column) and magnetic potential A1 ðt; x1 ; 0; 0Þ (right column) at time t ¼ 0:4. (–) ‘exact’ solutions; (+++) numerical solutions. (a) & (d) e ¼ 1, h ¼ 1 and Dt ¼ 0:2; (b) & (e) e ¼ 1=2, h ¼ 1=2 and Dt ¼ 0:05; (c) & (f) e ¼ 1=4, h ¼ 1=4 and Dt ¼ 0:0125; which corresponds to meshing strategy: h ¼ OðeÞ and Dt ¼ Oðe2 Þ.
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4.2. Applications Example 2. Exact results for plane wave solution of free MD, i.e. we choose e ¼ 1, d ¼ 12:97 in (3.1), (3.2). The external electromagnetic potentials are chosen as in (2.24), and the initial data is taken as in (2.20)– (2.22) with x1 ¼ 3; x2 ¼ x3 ¼ 5, V ð0Þ ¼ 1=p2 , V ð1Þ ¼ 1=2p3 , Að0Þ ¼ 0 and Að1Þ ¼ ð0; 1=7p2 ; 0ÞT . Thus the plane wave solution of free MD is given in (2.23)–(2.25) 3 We solve (3.1)–(3.5) on X ¼ ½p; p by our numerical method (3.27)–(3.29) with h ¼ p=8 and time step Dt ¼ 0:01. Fig. 2 shows the numerical results at different times. From Fig. 2, we can see that our method really gives exact results for plane wave solution of free MD. 0.04
x 10
0.03
–3
2
||ψ 3 ||
2
Real(ψ1(0,x2,0))
0.02 1.5
0.01
||ψ 2 ||
2
0 1
||ψ 1 ||
2
–0.01 –0.02
0.5
–0.03
||ψ || 2 4
–0.04
0 –3
–2
–1
(a)
0
x
1
2
3
0
0.2
0.4
0.6
(c)
2
0.8
1
t
0.04 0.1 0.03
V
Image(ψ1(0,x2,0))
0.02 0.05
0.01 0
A
A
3
1
0.01 0 –0.02 –0.03 –0.04
(b)
A
–3
–2
–1
0
x
2
1
2
3
–0.05
(d)
0
2
4
2
6
8
t
Fig. 2. Numerical results for Example 2 of wave function W1 ðt; 0; x2 ; 0Þ (left column) at t ¼ 1:0, time-evolution of position density and electromagnetic potentials (right column). (–) exact solutions; (+++) numerical solutions.
W. Bao, X.-G. Li / Journal of Computational Physics 199 (2004) 663–687
679
Fig. 3. Surface plots of the wave function jW1 ðt; x1 ; x2 ; 0Þj2 (left column) and electro potential V ðt; x1 ; x2 ; 0Þ (right column) at different times in Example 3 for Case 1. (a) t ¼ 0, (b) t ¼ 0:25, (c) t ¼ 0:5.
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0.26
0.45
0.258
0.4
0.256
0.35
0.254
4
0.3 2
0.252
||ψ || 1 ||ψ2||2 ||ψ ||2 3 ||ψ4||2
0.25 0.248
0.25 0.2 0.15
0.246
0.1
0.244 0.242
||ψ1||2 2 ||ψ2|| 2 ||ψ || 3 ||ψ ||2
0
(a)
0.5
1
1.5
2
0.05
(b)
0
0.5
1
1.5
2
t
Fig. 4. Time-evolution of position densities in Example 3. (a) Case 1; (b) Case 2.
Example 3. Dynamics of MD, i.e. we choose e ¼ 1, d ¼ 10:9149, V ext ðt; xÞ 0, Aext ðt; xÞ 0 in (3.1) and 2 2 2 ð1Þ (3.2), and Ak ðxÞ ¼ e4ðx1 þx2 þx3 Þ (1 6 k 6 3) in (3.4). We present results for two sets of parameters in (4.1): Case 1: c1 ¼ 3, c2 ¼ 4, c3 ¼ 5, c1 ¼ c2 ¼ c3 ¼ c4 ¼ 1, Case 2: c1 ¼ 2, c2 ¼ 4, c3 ¼ 8, c1 ¼ 8, c2 ¼ 4,c3 ¼ 4, c4 ¼ 1. 3 We solve this problem on a box ½8; 8 by our method with mesh size h ¼ 1=8 and time step Dt ¼ 0:002. 2 Fig. 3 shows the surfaces plots of jW1 ðt; x1 ; x2 ; 0Þj and V ðt; x1 ; x2 ; 0Þ at different times for Case 1. Fig. 4 2 shows time-evolution of particle densities kWj ðt; Þk ðj ¼ 1; 2; 3; 4Þ for Cases 1 and 2. 2 From Fig. 4, we can see that the total density kWk is conserved in the two cases. In case 1, the density for the first two components decreases for a period, attains their minimum, and then increases; where the time-evolution of the density for the other two components is in an opposite way in order to keep the conservation of the total density. Similar time-evolution pattern of density is formed in case 2 except more oscillation due to the nonuniform initial phase in the wave-function (cf. (4.1)). An interesting phenomenon in Fig. 4 is that after some time period, the density for each component almost keeps as a constant, i.e. there is no mass exchange between different components.
5. Conclusion An explicit, unconditionally stable and accurate time-splitting spectral method (TSSP) is designed for the Maxwell–Dirac system (MD). The method is explicit, unconditionally stable, time reversible, time transverse invariant, and of spectral-order accuracy in space and second-order accuracy in time. Moreover, it conserves the total position density exactly in discretized level and gives exact results for plane wave solution of free MD. Extensive numerical tests are presented to confirm the above properties of the numerical method. Our numerical tests also suggest the following meshing strategy (or e-resolution) is admissible in the ‘nonrelativistic’ limit regime (0 < e 1): spatial mesh size h ¼ OðeÞ and time step 4t ¼ Oðe2 Þ. The method is also applied to study dynamics of MD. In the future, we plan to use this state-of-the-art
W. Bao, X.-G. Li / Journal of Computational Physics 199 (2004) 663–687
681
numerical method to study more complicated time-evolution of fast (relativistic) electrons and positrons within external and self-generated electromagnetic fields.
Acknowledgements The authors acknowledge support by the National University of Singapore Grant No. R-151-000-027112 and thank very helpful discussions with Peter Markowich and Christof Sparber.
Appendix A. Diagnolize the matrix G nþ1=2 (x) in (3.19) and computation From (3.19), notice (3.18), we have 1 Gðtn ; xÞ þ 4Gðtnþ1=2 ; xÞ þ Gðtnþ1 ; xÞ 6 0 1 nþ1=2 nþ1=2 0 A3 ðxÞ A ðxÞ V nþ1=2 ðxÞ B C nþ1=2 nþ1=2 0 V nþ1=2 ðxÞ Aþ ðxÞ A3 ðxÞ C B ¼B C nþ1=2 @ A3nþ1=2 ðxÞ A A ðxÞ V nþ1=2 ðxÞ 0 nþ1=2 nþ1=2 nþ1=2 Aþ ðxÞ A3 ðxÞ 0 V ðxÞ
Gnþ1=2 ðxÞ ¼
ðA:1Þ
with nþ1=2
nþ1=2
nþ1=2
ðxÞ ¼ A1 ðxÞ iA2 ðxÞ; 1 V ðtn ; xÞ þ V ext ðtn ; xÞ þ 4ðV ðtnþ1=2 ; xÞ þ V ext ðtnþ1=2 ; xÞÞ þ V ðtnþ1 ; xÞ þ V ext ðtnþ1 ; xÞ ; V nþ1=2 ðxÞ ¼ 6
T nþ1=2 nþ1=2 nþ1=2 nþ1=2 ðxÞ ¼ A1 ðxÞ; A2 ðxÞ; A3 ðxÞ ; x 2 X; A A
nþ1=2
Ak
ðxÞ ¼
1 Ak ðtn ; xÞ þ Aext k ðtn ; xÞ þ 4ðAk ðtnþ1=2 ; xÞ 6 ext þ Ak ðtnþ1=2 ; xÞÞ þ Ak ðtnþ1 ; xÞ þ Aext k ðtnþ1 ; xÞ ;
k ¼ 1; 2; 3:
Since Gnþ1=2 ðxÞ is a U -matrix, it is diagonalizable. The characteristic polynomial of Gnþ1=2 ðxÞ is nþ1=2 k V nþ1=2 ðxÞ nþ1=2 0 A3 ðxÞ A ðxÞ nþ1=2 nþ1=2 0 k V nþ1=2 ðxÞ Aþ ðxÞ A3 ðxÞ det kI4 Gnþ1=2 ðxÞ ¼ nþ1=2 nþ1=2 A3 ðxÞ A ðxÞ k V nþ1=2 ðxÞ 0 nþ1=2 nþ1=2 Anþ1=2 ðxÞ A3 ðxÞ 0 kV ðxÞ þ h i2 2 2 ¼ k V nþ1=2 ðxÞ jAnþ1=2 ðxÞj ¼ 0: Thus the eigenvalues of Gnþ1=2 ðxÞ are nþ1=2 kþ ðxÞ;
with
nþ1=2 kþ ðxÞ;
nþ1=2 k ðxÞ;
nþ1=2 k ðxÞ
ðA:2Þ
682
W. Bao, X.-G. Li / Journal of Computational Physics 199 (2004) 663–687
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3 uX nþ1=2 nþ1=2 nþ1=2 nþ1=2 nþ1=2 k ðxÞ ¼ V ðxÞ jA ðxÞj ¼ V ðxÞ t jAj ðxÞj2 j¼1
and the corresponding eigenvectors are 0
nþ1=2 ðxÞ A
0
1
nþ1=2
A3
ðxÞ
1
B nþ1=2 C B C B Aþ ðxÞ C B A3nþ1=2 ðxÞ C nþ1=2 B C; B C ¼B ðxÞ ¼ B C C; v2 0 @ jAnþ1=2 ðxÞj A @ A jAnþ1=2 ðxÞj 0 1 0 0 1 0 jAnþ1=2 ðxÞj C B B jAnþ1=2 ðxÞj C C B 0 B C nþ1=2 nþ1=2 C; B v3 ; v ðxÞ ¼ B nþ1=2 ðxÞ ¼ C nþ1=2 4 B @ A ðxÞ A ðxÞ C A @ A3 nþ1=2 v1 ðxÞ
nþ1=2
A3
nþ1=2
ðxÞ
Aþ
ðxÞ
Let
nþ1=2 nþ1=2 nþ1=2 nþ1=2 Dnþ1=2 ðxÞ ¼ diag kþ ðxÞ; kþ ðxÞ; k ðxÞ; k ðxÞ ;
1 nþ1=2 nþ1=2 nþ1=2 nþ1=2 ðxÞ v2 ðxÞ v3 ðxÞ v4 ðxÞ : P nþ1=2 ðxÞ ¼ pffiffiffi nþ1=2 v1 2jA ðxÞj Thus Dnþ1=2 ðxÞ is a diagonal matrix, P nþ1=2 ðxÞ is a complex orthogonormal matrix, and they diagonalize the matrix Gnþ1=2 ðxÞ, i.e. T
Gnþ1=2 ðxÞ ¼ P nþ1=2 ðxÞDnþ1=2 ðxÞðP nþ1=2 ðxÞÞ ;
x 2 X:
ðA:3Þ
n , V ðtnþ1 ; xp;q;r Þ ¼ In order to compute Gnþ1=2 ðxp;q;r Þ (ðp; q; rÞ 2 M) used in (A.1), we need V ðtn ; xp;q;r Þ ¼ Vp;q;r n nþ1 nþ1 Vp;q;r , Aðtn ; xp;q;r Þ ¼ Ap;q;r , Aðtnþ1 ; xp;q;r Þ ¼ Ap;q;r , V ðtnþ1=2 ; xp;q;r Þ and Aðtnþ1=2 ; xp;q;r Þ. The first four terms are given in (3.27) and (3.28). The last two terms can be computed as following:
V ðtnþ1=2 ; xp;q;r Þ ¼
X
n Vej;k;l ðtnþ1=2 Þ eilj;k;l ðxp;q;r aÞ ;
ðj;k;lÞ2M
Aðtnþ1=2 ; xp;q;r Þ ¼
X
e n ðtnþ1=2 Þ eilj;k;l ðxp;q;r aÞ ; A j;k;l
ðj;k;lÞ2M
where for n ¼ 0:
0 Vej;k;l ðt1=2 Þ ¼
8 g 2 ð0Þ 2 Dt 2 g > ð0Þ Þ ð1Þ Þ > ðVg > j;k;l þ 2 ðV j;k;l þ ðjW j Þ j;k;l ðDtÞ =8e ; >
> > > g ð0Þ Þ < ðVg ðjWð0Þ j2 Þ =jlj;k;l j2 cosðDtjlj;k;l j=2eÞ j;k;l
> > > > > > > :
j ¼ k ¼ l ¼ 0;
j;k;l
e ð1Þ Þ þ ðVg j;k;l sinðDtjlj;k;l j=2eÞ jlj;k;l j g2 2 þ ðjWð0Þ j Þ j;k;l =jlj;k;l j
otherwise:
W. Bao, X.-G. Li / Journal of Computational Physics 199 (2004) 663–687
683
8 g g 2 g ð0Þ ð1Þ ð0Þ Dt 2 > j ¼ k ¼ l ¼ 0; > > ðA Þ j;k;l þ 2 ðA Þ j;k;l þ ðJ Þ j;k;l ðDtÞ =8e ; > > > h i > > 2 ð0Þ ð0Þ < ðAg Þ j;k;l ðJg Þ j;k;l =jlj;k;l j cosðDtjlj;k;l j=2eÞ 0 e A j;k;l ðt1=2 Þ ¼ > g > e > þ ðAð1Þ Þ j;k;l sinðDjlj;k;l j=2eÞ jlj;k;l > j > > > > : 2 g þ ðJð0Þ Þ j;k;l =jlj;k;l j otherwise: and for n > 0: 8 g2 2 n1 3 en > V ðt Þ 12 Vej;k;l ðtn1 Þ þ 3 ðjWn j Þ j;k;l ðDtÞ =8e2 ; > 2 j;k;l n > > >
> > > g n > Vej;k;l ðjWn j2 Þ j;k;l =jlj;k;l j2 cosðDtjlj;k;l j=2eÞ > > > > <
n g Vej;k;l ðtnþ1=2 Þ ¼ þ ð1 cosðjlj;k;l jDt=eÞÞ ðjWn j2 Þ j;k;l =jlj;k;l j2 > > > > i > > n1 n > > Vej;k;l ðtn1 Þ þ Vej;k;l ðtn Þ cosðjlj;k;l jDt=eÞ > > > > > g2 2 : 1 þ ðjWn j Þ j;k;l =jlj;k;l j 2 cosðjlj;k;l jDt=2eÞ 8 ðnÞ 3 en > e n1 ðtn1 Þ þ 3 ðJg Þ j;k;l ðDtÞ2 =8e2 ; A j;k;l ðtn Þ 12 A > j;k;l > 2 > > > h i > > 2 ðnÞ > A e n ðJg > Þ j;k;l =jlj;k;l j cosðDtjlj;k;l j=2eÞ > j;k;l > < h e n ðtnþ1=2 Þ ¼ 2 ðnÞ A Þ j;k;l =jlj;k;l j þ ð1 cosðjlj;k;l jDt=eÞÞ ðJg j;k;l > > i > > > e n1 ðtn1 Þ þ A e n ðtn Þ cosðjlj;k;l jDt=eÞ > A > j;k;l j;k;l > > > > > 2 ðnÞ 1 : þ ðJg Þ =jl j j;k;l
j;k;l
2 cosðjlj;k;l jDt=2eÞ
j ¼ k ¼ l ¼ 0;
otherwise: j ¼ k ¼ l ¼ 0;
otherwise:
The discretized current density JðnÞ is computed as ðnÞ ðnÞ Jp;q;r ¼ ðj1 Þp;q;r ; ðnÞ
ðnÞ
ðj2 Þp;q;r ;
ðjk Þp;q;r ¼ hWnp;q;r ; ak Wnp;q;r i;
ðnÞ
ðj3 Þp;q;r
T
;
k ¼ 1; 2; 3; n P 0 ðp; q; rÞ 2 N:
Appendix B. Diagnolize the matrix Mj;k;l in (3.24) From (3.24), notice (1.7), we have 0 B B ð1Þ ð2Þ ð3Þ Mj;k;l ¼ lj a1 þ lk a2 þ ll a3 þ e1 d1 ¼ B B @
e1 d1 0 ð3Þ ll ð1Þ
0 1 1 e d ð1Þ ð2Þ lj ilk ð2Þ
lj þ ilk
ð3Þ
ll
ð3Þ
ð1Þ
ð2Þ
ll ð1Þ ð2Þ lj þ ilk e1 d1
lj ilk ð3Þ ll 0
0
e1 d1
1 C C C: C A
684
W. Bao, X.-G. Li / Journal of Computational Physics 199 (2004) 663–687
The characteristic polynomial of Mj;k;l is ð3Þ k e1 d1 0 ll ð1Þ ð2Þ 0 k e1 d1 lj ilk det kI4 Mj;k;l ¼ ð3Þ ð1Þ ð2Þ ll lj þ ilk k þ e1 d1 ð1Þ ð2Þ ð3Þ l il ll 0 j k
2 2 ¼ k2 e2 d2 jlj;k;l j ¼ 0: Thus the eigenvalues of Mj;k;l are kj;k;l ;
kj;k;l ;
kj;k;l ;
kj;k;l
with kj;k;l ¼
ð1Þ
ð2Þ
lj þ ilk llð3Þ
0 k þ e1 d1
ðB:1Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 e2 d2 þ jlj;k;l j
and the corresponding eigenvectors are 0 1 0 1 0 kj;k;l þ e1 d1 B C B k þ e1 d1 C B C B j;k;l C 0 1 2 B C B C vj;k;l ¼ B ð3Þ C; vj;k;l ¼ B lð1Þ ilð2Þ C; l @ A @ j A l k ð1Þ ð2Þ ð3Þ lj þ ilk ll 1 1 0 0 ð3Þ ð1Þ ð2Þ ll lj þ ilk C B B ð2Þ C ð3Þ C C B B lð1Þ ll 3 4 j ilk C: C B B ; vj;k;l ¼ B vj;k;l ¼ B C 1 C 1 A @ @ kj;k;l þ e d A 0 kj;k;l þ e1 d1
0 Let
Dj;k;l ¼ diag kj;k;l ; kj;k;l ; kj;k;l ; kj;k;l ; 1 Pj;k;l ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v1j;k;l v2j;k;l 2ðk2j;k;l þ e1 d1 kj;k;l Þ
v3j;k;l
v4j;k;l :
Thus Dj;k;l is a diagonal matrix, Pj;k;l is a complex orthogonormal matrix, and they diagonalize the matrix Mj;k;l , i.e. Mj;k;l ¼ Pj;k;l Dj;k;l ðPj;k;l ÞT
ðj; k; lÞ 2 M:
ðB:2Þ
e n ðtnþ1 Þ and V e n ðtnþ1 Þ in (3.27) and (3.28) Appendix C. Computation of V j;k;l j;k;l For n ¼ 0:
0 Vej;k;l ðt1 Þ ¼
8 g 2 ð0Þ 2 2 g g > ð0Þ ð1Þ > > ðV Þ j;k;l þ Dt ðV Þ j;k;l þ ðjW j Þ j;k;l ðDtÞ =2e ; j ¼ k ¼ l ¼ 0; > > > > g2 2 ð0Þ Þ < ðVg ðjWð0Þ j Þ jlj;k;l j cosðDtjlj;k;l j=eÞ j;k;l
> > > > > > > :
j;k;l
e ð1Þ Þ þ ðVg j;k;l sinðDtjlj;k;l j=eÞ jlj;k;l j g þ ðjWð0Þ j2 Þ j;k;l =jlj;k;l j2
otherwise:
ðC:1Þ
W. Bao, X.-G. Li / Journal of Computational Physics 199 (2004) 663–687
e 0 ðt1 Þ ¼ A j;k;l
8 g g 2 ð0Þ > > Þ j;k;l ðDtÞ =2e2 ; j ¼ k ¼ l ¼ 0; ðAð0Þ Þ j;k;l þ Dt ðAð1Þ Þ j;k;l þ ðJg > > > h i > > 2 ð0Þ ð0Þ < ðAg Þ ðJg Þ =jl j cosðDtjl j=eÞ j;k;l
> > > > > > > :
j;k;l
j;k;l
j;k;l
g e þ ðAð1Þ Þ j;k;l sinðDtjlj;k;l j=eÞ jlj;k;l j 2 g ð0Þ þ ðJ Þ =jl j j;k;l
j;k;l
685
ðC:2Þ
otherwise:
and for n > 0: 8 g2 2 n n1 > ðtn Þ Vej;k;l ðtn1 Þ þ ðjWn j Þ j;k;l ðDtÞ =e2 ; 2 Vej;k;l > > >
< n g2 2 n Vej;k;l ðtnþ1 Þ ¼ 2 Vej;k;l ðjWn j Þ j;k;l =jlj;k;l j cosðDtjlj;k;l j=eÞ > > > > : g2 2 n1 ðtn1 Þ þ 2 ðjWn j Þ j;k;l =jlj;k;l j Vej;k;l 8 2 2 g ðnÞ en e n1 > > > 2 A j;k;l ðtn Þ A j;k;l ðtn1 Þ þ ðJ Þ j;k;l ðDtÞ =e ; < h i 2 e n ðtnþ1 Þ ¼ 2 A ðnÞ e n ðJg A Þ =jl j cosðDtjlj;k;l j=eÞ j;k;l j;k;l j;k;l j;k;l > > > : g n1 ðnÞ e ðtn1 Þ þ 2 ðJ Þ =jl j2 A j;k;l j;k;l j;k;l
j ¼ k ¼ l ¼ 0; ðC:3Þ otherwise: j ¼ k ¼ l ¼ 0; ðC:4Þ otherwise:
Appendix D. Proof of Theorem 3.1 Proof. From (3.29), notice (3.30), (3.20) and (3.25), Parseval’s equality, we have 2 X 1 2 nþ1 kWnþ1 kl2 ¼ Wp;q;r h1 h2 h3 ðp;q;rÞ2Q 2
X X iDt T g ilj;k;l ðxp;q;r aÞ Dj;k;l ðPj;k;l Þ ðW Þ j;k;l e ¼ Pj;k;l exp 2e ðp;q;rÞ2Q ðj;k;lÞ2M 2
X g Pj;k;l exp iDt Dj;k;l ðPj;k;l ÞT ðW ¼ M1 M2 M3 Þ j;k;l 2e ðj;k;lÞ2M X 2 g ¼ M1 M2 M3 Þ j;k;l ðW ðj;k;lÞ2M
2 X X X 1 ilj;k;l ðxp;q;r aÞ ¼ Wp;q;r e jW j2 ¼ M1 M2 M3 ðj;k;lÞ2M ðp;q;rÞ2Q ðp;q;rÞ2Q 2
X Dt nþ1=2 T nþ1=2 nþ1=2 ¼ ðxp;q;r Þ exp i D ðxp;q;r Þ ðP ðxp;q;r ÞÞ Wp;q;r P d ðp;q;rÞ2Q 2 X X 1 n 2 2 ¼ kWn kl2 ; n P 0: W p;q;r ¼ Wp;q;r ¼ h h h 1 2 3 ðp;q;rÞ2Q ðp;q;rÞ2Q Thus the equality (3.34) is obtained by induction.
ðD:1Þ
686
W. Bao, X.-G. Li / Journal of Computational Physics 199 (2004) 663–687
From (3.27), notice (C.1), (C.3), (3.30), we have g n n n1 Ve0;0;0 ðtnþ1 Þ ¼ 2 Ve0;0;0 ðtn Þ Ve0;0;0 ðtn1 Þ þ ðjWn j2 Þ 0;0;0 ðDtÞ2 =e2 n n1 ¼ 2 Ve0;0;0 ðtn Þ Ve0;0;0 ðtn1 Þ þ
2 X ðDtÞ jWð0Þ j2 M1 M2 M3 e2 ðp;q;rÞ2Q p;q;r
n1 n1 ¼ 2 Ve0;0;0 ðtn Þ Ve0;0;0 ðtn1 Þ þ
X ðDtÞ2 2 jWð0Þ j 2 M1 M2 M3 e ðp;q;rÞ2Q p;q;r
n2 n2 ¼ 3 Ve0;0;0 ðtn1 Þ 2 Ve0;0;0 ðtn2 Þ þ
ð1 þ 2ÞðDtÞ M1 M2 M3 e2
X
2
2 jWð0Þ p;q;r j :
ðD:2Þ
ðp;q;rÞ2Q
By induction, we get nðn þ 1ÞðDtÞ n 0 0 Ve0;0;0 ðtnþ1 Þ ¼ ðn þ 1Þ Ve0;0;0 ðt1 Þ n Ve0;0;0 ðt0 Þ þ 2M1 M2 M3 e2 ð0Þ ð1Þ ¼ Ve0;0;0 ðt0 Þ þ tnþ1 Ve0;0;0 ðt0 Þ þ
X
2 tnþ1
2M1 M2 M3
2
e2
X
ð0Þ jWp;q;r j
2
ðp;q;rÞ2Q 2
jWð0Þ p;q;r j ;
ðD:3Þ
n P 0:
ðp;q;rÞ2Q
From (3.27), notice (D.3), (3.30) and (3.32), we get X X X 1 nþ1 n Vej;k;l DMeanðV nþ1 Þ ¼ Vp;q;r ¼ ðtnþ1 Þeilj;k;l ðxp;q;r aÞ h1 h2 h3 ðp;q;rÞ2Q ðp;q;rÞ2Q ðj;k;lÞ2M X X n n e V j;k;l ðtnþ1 Þ ¼ eilj;k;l ðxp;q;r aÞ ¼ M1 M2 M3 Ve0;0;0 ðtnþ1 Þ ðj;k;lÞ2M
ðp;q;rÞ2Q
h
i t2 X ð1Þ 2 0 ðt0 Þ þ tnþ1 Ve0;0;0 ðt0 Þ þ nþ12 jWð0Þ j ¼ M1 M2 M3 Ve0;0;0 2e ðp;q;rÞ2Q p;q;r ¼
X ðp;q;rÞ2Q
ð0Þ Vp;q;r þ tnþ1
X ðp;q;rÞ2Q
ð1Þ Vp;q;r þ
2 X tnþ1 jWð0Þ j2 ; 2e2 ðp;q;rÞ2Q p;q;r
Thus the desired equality (3.35) is a combination of (3.32) and (D.4).
n P 0:
ðD:4Þ
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