A simple pseudospectral method for the

A simple pseudospectral method for the computation of the time-dependent Dirac equation with Perfectly Matched Layers. Application to quantum relativistic laser physics. X. Antoine Institut Elie Cartan de Lorraine, Université de Lorraine, Sphinx team, Inria Nancy-Grand Est, F-54506 Vandoeuvre-lès-Nancy Cedex, France∗

E. Lorin Centre de Recherches Mathématiques, Université de Montréal, Pavillon André-Aisenstadt, 2920 Chemin de la tour, Montréal (Québec), H3T 1J4. School of Mathematics and Statistics, Carleton University, Ottawa, Canada, K1S 5B6† (Dated: December 29, 2018) A simple time-splitting pseudospectral method for the computation of the Dirac equation with Perfectly Matched Layers (PML) is proposed. Within this approach, basic and widely used FFTbased solvers can be adapted without much effort to compute Initial Boundary Value Problems (IBVP) for the time-dependent Dirac equation with absorbing boundary layers. Some numerical examples from laser-physics are proposed to illustrate the method.

1.

INTRODUCTION

This paper is concerned with the numerical computation of the time-dependent Dirac equation, especially for physical problems involving delocalized wavefunctions. More specifically, we are interested in the numerical solution to the Dirac equation on a truncated domain with absorbing boundary layers, hence limiting the periodic conditions effects. In this goal, we then propose a simple combination of a pseudospectral method with Perfectly Matched Layers (PMLs), allowing to consider delocalized wavefunctions, as for instance observed when quantum relativistic particles are subject to strong fields. Thanks to a relatively new and simple Fourier-based discretization of spatial differential operators, it is possible to impose PML for solving the Dirac equation on a bounded domain. Although, overall, the Fourier-based method still imposes periodic boundary conditions, the outgoing/incoming waves are in fact mainly absorbed. We think that due to the simplicity of the proposed method, most of existing Fourier-based codes could easily be modified to include the proposed methodology. The Dirac equation is a relativistic wave equation which has gain much attention these past 15 years due to the development of 2-d material, such as graphene [1–3], intenselaser-molecule interaction [4–7], in particular for pair production [8–10], or from heavy ion collisions modeling and simulation for quark-antiquark production [11–13]. In the same time, there has been a tremendous progress in the development of efficient and accurate computational methods for the solution to the Dirac equation [6, 14– 26], modeling in particular molecules subject to intense external laser fields. In this spirit, we have developed [8], a Schwinger-like pair production procedure from the

∗ †

[email protected] [email protected]

interaction of intense laser fields with heavy molecules occuring at specific resonances [8, 27]. In this paper, we are more specificially interested in the interaction of relativistic interaction of atoms, molecules or wavepackets with intense and short laser pulses. The key point is that the laser field actually delocalizes the wavefunction, and the latter can actually interact with the domain boundary. In order to avoid artificial reflection it is necessary to impose absorbing boundary conditions [28], absorbing complex potentials or perfectly matched layers [29]. Theoretically, this approach allows to benefit from the spectral convergence and simplicity of Fourier-based methods, more generally pseudospectral methods, on bounded domains reducing the periodic boundary condition effect thanks to artificial wave absorption at the domain boundary. Perfectly Matched Layers are now widely used in many engineering and physics simulations codes [30–43] to model exterior domains and to avoid any unphysical reflection at the boundary. PML for the Dirac equation were developed in [29] and approximated with a real space method. The derivation of high-order absorbing boundary conditions (ABCs) for the Dirac equation were proposed in [28]. We also refer to [44] for an overview of PMLs and ABCs for quantum wave equations including the Dirac equation and to [14, 15, 17–19, 21, 24, 45–51] for different approaches for solving the Dirac equation in real or Fourier space. The combination of pseudospectral methods and PMLs is possible thanks to the following simple and original idea. For example in the x-direction, let us denote by ξx the dual Fourier variable and by Fx (resp. Fx−1 ) the Fourier (resp. inverse Fourier) transform in x. We consider a as a given x-dependent function. Then, for any function f , it is possible to formally rewrite a(x)∂x f (x) as Fx−1 a(x)iξx F(f )(ξx ) (x) on an unbounded domain (pseudodifferential operator representation [52]). Let us remark that the latter is a real space function, although

2 the derivative is approximated using the Fourier transform, and the function (x, ξx ) 7→ ia(x)ξx is nothing but the symbol of a(x)∂x . In practice the function a involves the stretching coordinates function modeling the PML on a bounded domain allowing real space non-reflecting conditions at the domain boundary. The time-dependent Dirac equation under consideration reads [53] i∂t ψ(t, x) = Hψ(t, x),

(1)

where ψ(t, x) is the time and coordinate dependent fourspinor, and H is the Hamiltonian operator. The latter is given by H = α · [cp − eA(t, x)] + βmc2 + I4 V (t, x),

(2)

where the momentum operator is p = −i∇. More specifically, the Dirac equation that we consider reads [53]

i∂t ψ(t, x) =

αx −ic∂x − eAx (t, x) +αy −ic∂y − eAy (t, x) +αz −ic∂z − eAz (t, x) + βmc2 +I4 V (t, x) ψ(t, x),

where ψ(t, x) ∈ L2 (R3 ) ⊗ C4 is the time and coordinate (x = (x, y, z)) dependent four-spinor. In (3), A(t, x) represents the three space components of the electromagnetic vector potential, V (t, x) = eA0 (t, x) + Vnuc. (x) is the sum of the scalar and interaction potentials, e is the electric charge (with e = −|e| for an electron), I4 is the 4 × 4 unit matrix and α = (αν )ν=x,y,z , β are the Dirac matrices. In this work, the Dirac representation is used, where 0 σν I 0 αν = , β= 2 . (3) σν 0 0 −I2 The σν are the usual 2 × 2 Pauli matrices defined as 0 1 0 −i 1 0 σx = , σy = and σz = , (4) 1 0 i 0 0 −1 while I2 is the 2 × 2 unit matrix. Note that the light velocity c and fermion mass m are kept explicit in Eq. (2), allowing to adapt the method easily to natural or atomic units (a.u.). The paper is organized as follows. We recall the basics of PMLs for the Dirac equation in Section 2. Section 3 is dedicated to the derivation of the pseudospectral approximation for the Initial Boundary Value Problem (IBVP) time-dependent Dirac equation on a bounded domain with PML. Numerical experiments are presented in Section 4. We finally conclude in Section 5.

2.

PML FOR THE DIRAC EQUATION

The time-dependent Dirac equation is then considered on a bounded (truncated) physical domain denoted by DPhy . We add a layer, that is called DPML , surrounding DPhy , and stretch the coordinates in all the directions. The overall computational domain is next defined by: D = DPhy ∪ DPML . We refer to [44] for the construction of PMLs for quantum wave equations and more specifically to [29] for the derivation and analysis of PMLs for the Dirac equation. The starting point is the stretching of the real coordinates in the complex plane. In [29], the author uses the following variables Z i ν νe = ν + σ(s)ds, ω L∗ν where ω is the dual Fourier variable to t and the so-called absorbing function σ is such that σ(ν) = 0 if s < L∗ν , for some imposed L∗ν < Lν . Then, the partial derivative ∂ν , with ν = x, y, z, is shown to be formally transformed into ∂ν ψ(t, ·) →

∂t ∂ν ψ(t, ·) ∂t + σ eZν (ν) (5) 1 ωeiωt = ∂ν Ft ψ(ω, ·)dω, 2π R ω − ie σν (ν)

where Ft (resp. Fν ) denotes the Fourier transform with respect to t (resp. ν) and σ(|ν| − Lν ), L∗ν 6 |ν| < Lν , σ eν (ν) = 0, |ν| < L∗ν . Let us remark that the transformation (5) can also be formally rewritten as ∂ν ψ(t, ·) →

1 1+σ eν (ν)∂t−1

∂ν ψ(t, ·).

In the following, we will rather consider the following change of variables [43] involving only the space variable Z ν iθ νe = ν + e σ(s)ds, L∗ ν

where θ ∈ (0, π/2). We then define Sν (ν) := 1 + eiθν σ e(ν), with ν = x, y, z. In the following, the partial derivatives are transformed into ∂ν →

1 1 ∂ν = ∂ν , iθ Sν (ν) 1 + e νσ eν (ν)

(6)

where σ e vanishes in DPhy and Sν is equal to 1. From now on, let us consider the transformation (6), and the associated new Hamiltonian HPML = α · [cT · p − eA(t, x)] + βmc2 + I4 V (t, x),(7)

3 where T = (Sx−1 (x), Sy−1 (y), Sz−1 (z))T . Several types of functions can be selected. An exhaustive study of the absorbing functions σ is proposed in [54] for Schrödinger equations Type I: σ0 (ν + δν )2 , Type II: σ0 (ν + δν )3 , Type III: − σ0 /ν, Type IV: σ0 /ν 2 , Type V: − σ0 /ν − σ0 /δν , Type VI: σ0 /ν 2 − σ0 /δν2 , for ν = x, y, z. The main difficulty from the pseudospectral point of view is the space-dependence of the coefficients Sν−1 which prevents the direct application of the Fourier transform on the equation. A simple trick will however allow for combining the efficiency of the pseudospectral method and the computation of the nonconstant coefficient Dirac equation.

3.

PSEUDOSPECTRAL-PML (PS-PML) METHOD

A splitting of the Dirac Hamiltonian is used below. Although operator splitting is not fundamentally required in the proposed methodology, it allows to simplify the implementation of the method while keeping a good accuracy. Based on (2), we define the operators A= B= C= D=

−icSx−1 (x)αx ∂x , −icSy−1 (y)αy ∂y , −icSz−1 (z)αz ∂z , βmc2 + I4 V (t, x)

(8) (9) − eα · A(t, x).

(10) (11)

From time tn to tn+1 , we then successively solve [24] (the x-dependence in the wavefunction argument for notational convenience) i∂t ψ (1) (t) = Aψ (1) (t), ψ (1) (tn ) = ψ n ,

t ∈ [tn , tn(12) 1)

i∂t ψ (2) (t) = Bψ (2) (t), ψ (2) (tn ) = ψ (1) (tn1 ), t ∈ [tn , tn(13) ) 2

and we use the same notation as nalize αν = Πν Λν Π†ν , where  1 0 0 0 1 0 Λν =  0 0 −1 0 0 0

in [55]. We first diago 0 0  . 0  −1

T

and Π†ν = Πν . The matrices Πν are defined as follows   0 1 1 0 1  1 0 0 −1  Πx = √  , 2 1 0 0 1 0 1 −1 0   0 −i −i 0 1  1 0 0 1 Πy = √  , 2 −i 0 0 i 0 1 −1 0   1 0 0 −1 1  0 −1 −1 0  Πz = √  . 2 1 0 0 1 0 1 −1 0 We set φ := Π†ν ψ, which then satisfies i∂t φ(t) = −icΛν Sν−1 (ν)∂ν φ(t), φ(tn ) = Π†ν ψ n , t ∈ [tn , tn+1 ). We denote the set of grid-points by DNx ,Ny ,Nz = xk1 ,k2 ,k3 = (xk1 , yk2 , zk3 ) (k1 ,k2 ,k3 )∈O

Nx Ny Nz

with ONx Ny Nz = (k1 , k2 , k3 ) ∈ N3 / k1 = 0, · · · , Nx − 1; k2 = 0, · · · , Ny − 1; k3 = 0, · · · , Nz − 1 . Then, let us introduce the following mesh sizes xk1 +1 − xk1 = hx = 2ax /Nx , yk2 +1 − yk2 = hy = 2ay /Ny , zk3 +1 − zk3 = hz = 2az /Nz .

(17)

The corresponding discrete wavenumbers are defined by ξp,q,r := (ξp , ξq , ξr ), where ξp = pπ/ax with p ∈ {−Nx /2, · · · , Nx /2 − 1}, ξq = qπ/ay with q ∈ (4) (4) (4) (3) i∂t ψ (t) = Dψ (t), ψ (tn ) = ψ (tn3 ), t ∈ [tn , tn+1 (15)) {−Ny /2, · · · , Ny /2 − 1} and ξr = rπ/az with r ∈ and ψ n+1 = ψ (4) (tn+1 ), (16) {−Nz /2, · · · , Nz /2 − 1}. In the sequel of the paper, we denote by φ(`) , with ` ∈ {1, 2, 3, 4}, the `th component where tni − tn = ∆t for i ∈ {1, · · · , 3}. In the next subof the spectral approximation of φ = Π†ν ψ. We also use (`) (`) sections, we detail the space-time approximation of equathe notation φk1 (t, y, z) = φ(`) (t, xk1 , y, z), φk2 (t, x, z) = tions (12)-(15). Cylindrical coordinates could be used as (`) φ(`) (t, x, yk2 , z) and φk3 (t, x, y) = φe(`) (t, x, y, zk3 ). The well (see e.g. [21]). partial Fourier coefficients are such that  NX x −1   b(`) (`)  φ (t, y, z) = φk1 (t, y, z)e−iξp (xk1 +ax ) ,  p A. Pseudospectral approximation in space    k1 =0    Ny −1  X (`) We consider for convenience the 3-dimensional system φb(`) φk2 (t, x, z)e−iξq (yk2 +ay ) , q (t, x, z) =  in cartesian coordinates, for ν = x, y, z in the domain  k2 =0    D = [−ax , ax , ] × [−ay , ay ] × [−az , az ], NX z −1   (`)  (`) b  φ (t, x, y) = φk3 (t, x, y)e−iξr (zk3 +az ) .   r −1 n i∂t ψ(t) = −icαν Sν (ν)∂ν ψ(t), ψ(tn ) = ψ , t ∈ [tn , tn+1 ) k3 =0 i∂t ψ (3) (t) = Cψ (3) (t), ψ (3) (tn ) = ψ (2) (tn2 ), t ∈ [tn , tn(14) ) 3

4 We can then introduce the inverse partial Fourier pseudospectral approximations, in the x-, y- and z-directions, respectively,  Nx /2−1 X   iξp (xk1 +ax )  φe(`) (t, y, z) = 1  φb(`) , p (t, y, z)e  k 1  N x   p=−N /2 x    Ny /2−1  X 1 (`) iξq (yk2 +ay ) φb(`) , φek2 (t, x, z) = q (t, x, z)e  N y  q=−N /2  y   Nz /2−1   X 1  (`)  iξr (zk3 +az ) e  φ (t, x, y) = φb(`) .  r (t, x, y)e  k3 Nz

where [[·]] is defined in (18). We then deduce ψhn1 = Πx φnh1 . This gives an approximation of Eq. (12), where the corresponding operator is denoted (x) by Ph (∆t). 2. Second step: integration of the generalized transport equation in Fourier space in the y-direction ∂t φ + cSy−1 (y)Λy ∂y φ = 0. We set χnh1 := Π†y ψhn1 and we approximately solve (13) by using the following scheme

r=−Nz /2

We finally define the approximate first-order partial derivatives such that for any ν ∂ν φ(`) (tn , xk1 ,k2 ,k3 ) ≈ [[∂ν ]]φ(`) k1 ,k2 ,k3 and

[[∂x ]]φ(`) [[∂y ]]φ(`) [[∂z ]]φ(`)

k1 ,k2 ,k3

k1 ,k2 ,k3

k1 ,k2 ,k3

:=

:=

:=

1 Nx 1 Ny 1 Nz

Nx /2−1

X p=−Nx /2 Ny /2−1

X q=−Ny /2 Nz /2−1

X

b(`) iξp φek2 ,k3 p eiξp (xk1 +ax ) , b(`) y) iξq φek1 ,k3 q eiξq (xk2 +a(18) , b(`) iξr φek1 ,k2 r eiξr (xk3 +az ) .

φnh2 = χnh1 − c∆tSy−1 (yh )Λy [[∂y ]]χnh2 . We then obtain ψhn2 = Πy φnh2 . 3. Third step: setting χnh2 := Π†z ψhn2 , we apply the same procedure in the z-direction, and compute ψhn3 . The corresponding operator is denoted by (z) Ph (∆t). 4. Fourth step: integration of the source from tn to tn+1 Z ψhn+1 = T exp −i

tn+1

tn

Z × exp −ie

r=−Nz /2

tn+1

tn

In the following, the index h will be used (e.g. in φnh = {φnk1 ,k2 ,k3 }k1 ,k2 ,k3 ) to denote a spectral approximation to a given wavefunction (e.g. φn ). This discretization not only allows to select the spatial step as large as wanted, but it also preserves the very high spatial accuracy, the parallel computing structure and the scalability of the split method developed in [19].

B.

Time-Splitting PSeudospectral method with PML (TSSP-PML)

As proposed before, we use a splitting of the TDDE into four time-dependent systems (8), (9), (10) and (11). We denote by xh = (xh , yh , zh ) the nodes of a real-space grid. At iteration n, the approximate wave function is denoted by ψhn . From a stability point of view, an implicit scheme should be implemented. The TSSP (TimeSplitting pseudoSPectral) algorithm then reads 1. First step: integration of the generalized transport equation in Fourier space in the x-direction. One sets φnh := Π†x ψhn , and the system ∂t φ + cSx−1 (x)Λx ∂x φ = 0,

φ(tn , ·) = Π†x ψ(tn , ·),

is approximately solved by φnh1

=

φnh

−

c∆tSx−1 (xh )Λx [[∂x ]]φnh1 ,

dτ βmc2 − eα · Ah (τ ) dτ Vh (τ ) ψhn3 ,

(19)

where Ah is the approximate electric potential. The corresponding operator is denoted by Qnh (∆t). This splitting scheme is implicit, and can be compactly expressed as a composition of the four operators (z)

(y)

(x)

ψhn+1 = Qnh (∆t)Ph (∆t)Ph (∆t)Ph (∆t)ψhn . In practice, a second-order splitting scheme should be implemented, i.e. (y) ∆t (x) ∆t (z) ∆t (x) ∆t Ph Ph Ph Qnh (∆t) ψhn+1 = Ph 4 2 4 2 (20) (z) ∆t (x) ∆t (y) ∆t (x) ∆t Ph Ph Ph ψhn . Ph 2 4 2 4 Thanks to the operator [[∂ν ]], it is possible to implement the real-space approximation of the overall wave function on a bounded domain imposing in particular PMLs, resulting in the TSSP-PML scheme. Remark 3.1 From a pratical point of view, we proceed as follows in order to avoid the numerical solution to linear systems. Say for the First step, we can simply instead solve ∂t φ + cSx−1 (x)Λx ∂x φ = 0, φ(tn , ·) = Π†x ψ(tn , ·), (21) by using φnh1 :=

1 1 n ψhn1 + 1 − φ , Sx (xh ) Sx (xh ) h

5 where ψhn1 is solution to ψhn1

= φnh − c∆tΛx [[∂x ]]φnh .

In other words, φnh1 is an approximation to φ(tn1 , x) =

1 Fx−1 1 − ic∆tξx Fx (φ)(tn1 , ξx ) Sx (x) (22) 1 φ(tn , x). + 1− Sx (x)

This approximation is justified as follows. From time tn to tn1 , we integrate (21) and get φ(tn1 , x) = φ(tn , x) −

c∆t −1 Fx iξx Fx (φ)(tn , ξ) Sx (x)

+O(∆t2 ) 1 (23) Fx−1 1 − ic∆tξx Fx (φ)(tn , ξ) = Sx (x) 1 φ(tn , x) + O(∆t2 ) + 1− Sx (x) As a consequence, (22) is consistent with (23). 4.

NUMERICAL SIMULATIONS

We propose some numerical experiments in 2-d, illustrating the efficiency of the combined method PMLTSSP method with a second-order operator splitting. We are mainly interested in the evolution of wavepacket either subject to a static potential or of an external field. In our simulations, we have used several libraries which are listed below. A.

Technical details about the code and parallel computing aspects

The Fourier transforms are performed with the sequential and parallel version 3.3.4 of fftw. Standard linear algebra libraries gsl (version 1.9) and blas were also used. Finally openMPI 1.6.3 was used for message passing with non-blocking communication. The code was implemented in C++ and the compiler is the version 4.7.0 of gcc. The tests are performed by using a C++-code with MPI-library. The method is implemented on the cluser mammouth-parallel II from the RQCHP. The total processing power of this machine is 333 400 GFlops, and possesses 39648 cores: 3216 processors AMD Opteron 12 cores at 2.1 GHz, and 88 processors AMD Opteron 12 cores 2.2 GHz. The total memory is 57.6 TB and the computer-networking communications is Infiniband QDR (4 GB/sec). Regarding the communication between processes, by default mammouth II will select the closest processors in the same node, then the processors from the nodes in the same topological ring, and finally through different switch-levels. Notice that for a low number of processors, the communication between the processes within the same node is done thanks

to a shared memory. The detailed of the Dirac equation solver without PML is presented in detailed in [14], Section 4. Let us recall the main features of the code in 2-d. In the (x, z) coordinates, we proceed by alternating the directions. We first decompose the domain by layers in the z-direction, then • We successively perform the evolution (FFT) sequentially in the x-direction, and by layer in the z-direction. Each processor manages one layer in z. A perfect scaling for this step is expected as it does not require any transmission between nodes. • For all x, we perform the evolution in the zdirection using the parallel FFT (fftw). The performance of this step is then fully dependent on the parallelization of the one-dimensional FFT. Notice that the presence of the PML does not deteriorate the efficiency of the overall method compared to usual FFT-methods. In particular, the computational complexity is simple to established, and is given by C after NT time iterations C = O NTf Nx Nz log(Nx Nz ) . The scalability and performance of the TSSP method without PML is fully studied in [19], where in particular it is shown a good speed-up of this method.

B.

Numerical experiments

The second-order TSSP-PML method (20) is implemented, where the discrete operators [[·]] given by (18) are used in the directions ν = x, z. We then choose Sν as follows, for ν = x, z, 1, |ν| < L∗ν , (24) Sν (ν) = iθ ∗ 1 + e σ(|ν| − Lv ), Lν 6 |ν| < Lν . Test 1 : wavepacket subject to an external laser field. In this first test, we consider a wavepacket ψ(x, y, z, 0) = N [1, 0, 0, 0]T exp − (x2 + y 2 + z 2 )/∆2 (25) × exp(ikx x), subject to an external potential and propagating with a fixed wavenumber k = (5, 0), and ∆ = 128 in (25). The time step is given by ∆t = 4.56 × 10−4 . The physical domain is D = [−8, 8]2 and is discretized with Nx × Nz = 2562 grid points. We impose a linearly polarized electric field such that Ax is identically null, and Az (t, x, z) is as in (27), with A0 = 100/137, ω = 100, n = 1, n0 = 19 and Tf = 1.824. As an example, we report in Fig. 1 the electric potential Az at time T = 0.456 and 1.824 and the initial data. In Figs. P 2-5, we compare the 4 evolution of the density d(T, x, z) = i=1 |ψi (T, x, z)|2 as a function of time, with i) periodic boundary conditions, with PML of ii) type III and iii) type IV, with σ0 = 10−2 , θ = π/4 as well as iv) type V and VI with δν = Lν − L∗ν .

6

FIG. 1. Test 1 : initial density (Top), Az (T, x, y) at times t = 0.456 (Middle) and 1.824 (Bottom).

The snapshots correspond to times t = 0.456, 1.14, 1.368 and T = 1.824, for respectively the time iterations 1000, 2500, 3000 and 4000. The best absorption is obtained with the Type-VI absorbing function, although the results are relatively close with all the absorbing functions. We report in Fig. 6, the maximum of the density (in logscale) as a function of the time iterations for 6 different absorbing functions, showing that they all provide relatively good properties for truncating the computational domain.

Test 2 : 3-nucleus system. In this test, a Gaussian wavepacket centered at (0, 0) Fig. 7 (Left), is injected in

FIG. 2. Test 1 : density at time t = 0.456 (first line) with periodic boundary conditions (without PML). The second (respectively third) line reports the results for the density with TSSP-PML, with a PML of Type-III (respectively Type-IV), setting σ0 = 10−2 , θ = π/4. The fourth (respectively fifth) line gives the same kind of results but for the Type-V (respectively Type-VI) PML, with σ0 = 10−2 , θ = π/4 and δν = Lν − L∗ν .

7



8 10 0

10 -1

10 -2

1000

1500

2000

2500

3000

3500

4000

FIG. 6. Test 1 : maximum of the density in logscale, as a function of the time iteration, for 6 different types of PML.

a 3-nucleus interaction potential Fig 7 (Middle) ZA V (x, z) = − p 2 (x − xA ) + (z − zA )2 + 2 ZB −p 2 (x − xB ) + (z − zB )2 + 2 ZC −p (x − xC )2 + (z − zC )2 + 2

(26)

with ZA = ZB = ZC = 10a.u., (xA , zA ) = (−2, 0), (xB , zB ) = (2, 0) and (xC , zC ) = (0, −2). The overall computational domain is D = [−6, 6]2 , with Nx × Ny = 1282 grid points and ∆t = 3 × 10−3 . Notice that this time step does not allow for the capture of very small scale effects, such as the zitterbewegung. As we are here interested in larger scales effects, more specifically the absorption layer effect, we allow a larger time step. For practical application, we however suggest to use, as usual, a time step 6 1/2mc2 to get a better precision. We report on Fig. 7 (Bottom) the total density at time T = 3 × 10−1 a.u., i.e. d(T, x, z), for P4 (x, z) ∈ D, where d(T, x, z) = i=1 |ψi (T, x, z)|2 , setting ψ = (ψ1 , ψ2 , ψ3 , ψ4 ) as the Dirac wave function. Without external excitation the wavepacket is then trapped by the potential. We then plug an external laser field which drives the wavepacket: a linearly polarized electric field is imposed such that Ax is identically null, and Az (t, x, z) is defined as follows Az (t, x, z) = A0 cos ω(x/c − t) f ω(x/c − t) (27)

FIG. 5. Test 1 : density at time t = T = 1.824 (first line) with periodic boundary conditions (without PML). The second (respectively third) line reports the results for the density with TSSP-PML, with a PML of Type-III (respectively TypeIV), setting σ0 = 10−2 , θ = π/4. The fourth (respectively fifth) line gives the same kind of results but for the Type-V (respectively Type-VI) PML, with σ0 = 10−2 , θ = π/4 and δν = Lν − L∗ν .

with A0 = 1000/137, ω = 1, and where the envelope function is linear, then constant, and finally linear. More specifically f is defined as  ω   t ∈ [0, nπ/ω],   nπ t, 1, t ∈ [nπ/ω, (n + n0 )π/ω], f (t) =  0  ω (2n + n )π   t− , t ∈ [(n + n0 )π/ω, (2n + n0 )π/ω], nπ ω

9

FIG. 8. Test 2 : z-component of the external laser potential: Az (T, x, z) for T = 3 × 10−1 a.u.

FIG. 7. Test 2 : (Top) Initial density. (Middle) 3-nucleus potential. (Bottom) Density at time T = 3×10−1 a.u. without external laser field.

where 2n + n0 denotes the total number of half-cycles. In the test, we took n = n0 = 1. We report the zcomponent of the external laser potential in Fig. 8 at final time. We also include a non-null wavenumber in the initial wavepacket, k = (10, 0), out of the ions influence. The wavepacket will eventually be absorbed by the PML unlike the solution without PML. At times t = 3 × 10−2 , t = 6 × 10−2 a.u., t = 3 × 10−1 and t = 3.6 × 10−1 , we compare 3 solutions Fig. 9, 10, 11, 12: 1) the solution of reference computed on [−18, 18]×[−18, 18], 2) the solution without PML, and 3) the solution with Type V-PML and σ0 = 10−2 , θ = π/4, and δ = 1.2. At time t = 6 × 10−2 , the 3 solutions look identical, as the wavefunction has not reached the

boundary of the computational domain [−6, 6] × [−6, 6]. At t = T = 3 × 10−1 , we however clearly see that the wavefunction of reference has mainly left the region [−6, 6] × [−6, 6], while the PML solution was almost totally absorbed and the solution without PML, was maintained in the computational domain due to nonabsorbing boundary layers. This is also illustrated on Fig. 13, where the `2 -norm as function of time in the D is represented for the 3 solutions. We see that, unlike the solution without PML, the PML-solution makes decrease the `2 − norm, thanks to the absorbing layers. We also report in Fig. 13 (Top), we represent the solution of reference at time T = 0.3a.u. in a bigger domain [−18, 18] × [−18, 18]. By comparison, we see that on this example, the solution of reference has also a decreasing `2 − norm in the zone D ( t, kdRef (t, ·)k 2 ), see Fig. 13 |D 2 where the ` -norm is represented in logscale as a function of time (Bottom). We also report the solution of reference in the domain [−18.18] × [−18, 18] at time T = 0.36a.u. Test 3 : evolution of wavepacket subject to a repulsive weakly nonlinear potential. We consider the two-dimensional Dirac equation with a nonlinear potential V (x, z, ψ) = − p −p

ZA (x − xA

)2

+ (z − zA )2 + 1 ZB

(x − xB )2 + (z − zB )2 + 1

(28) + |ψ|2 ,

with ZA = ZB = 2, (xA , zA ) = (−1, 4) and (xB , zB ) = (1, 4). The initial density with wave vector k = (2, 10) is represented in Fig. 14. More specifically, the initial data is given by ψ(x, y, z, 0) = N [1, 0, 0, 0]T exp − (x2 + y 2 + z 2 )/∆2 (29) × exp(ikx x),

10

FIG. 9. Test 2 : From top to bottom: No PML, reference and Type V-PML solutions at time t = 3 × 10−2 a.u..


where ∆ = 128 in (29) and N is a normalization coefficient, such that kψ(·, 0)k(L2 (D))4 = 1. The overall computational domain is D = [−8, 8]2 , with Nx × Ny = 2562 grid points and ∆t = 4.56 × 10−4 . We report on Fig. 15 the total density at time T = 0.912 a.u. in logscale, i.e. log(d(T, x, z), for (x, z) ∈ D, where d(T, x, z) = P4 2 i=1 |ψi (T, x, z)| , setting ψ = (ψ1 , ψ2 , ψ3 , ψ4 ) as the Dirac wave function. At that time, the wavepacket passed through the boundary of the computational domain at z = ±8. The nonlinearity is numerically treated explicitly. The use of the logscale allows to fairly report the accuracy of the PML. In (24), we take Lν = 8, L∗ν = 0.8Lν , θ = π/4 (similarly to the Schrödinger equation [43, 54]), and we pick the Type-IV PML, i.e. σ : ν 7→ σ0 /ν 2 (singular profile [54]). We represent the so-

lution with different values of σ0 . These results show that the PML properly absorbs except for σ0 = 5, with reflection magnitude as weak as 10−6 for σ0 = 10−2 . Let us remark that other PMLs could be considered as well [29], but we intend here to prove the feasibility of the method for various PMLs. For completeness and by comparisons with results obtained in Fig. 15, we also report in Fig. 16 the solution to the Dirac equation in D with zero Dirichlet boundary condition by using the real-space quantum lattice Bolztmann method proposed in [19] and illustrating the total wave reflection at the domain boundary when PMLs or ABCs are not used. This simple example illustrates the fact without using PML and or low-order ABC, a wavefunction is totally reflected. This is a standard issue when solving the Dirac equation, in particular

11


when studying laser-molecule interaction. We report in Fig. 19 (Top), the maximum of the density as a function of time for σ0 = 10−3 , 10−2 , 10−1 , 1, 5 in logscale, with the PML of Type IV for θ = π/4. We clearly see that σ0 = 10−2 provides the best absorption properties. We next compare in Fig. 17, for σ0 = 10−2 , the reflection for θ = π/16, θ = π/8, θ = π/4 and θ = π/2 showing the importance of properly selecting θ, with best absorption at θ = π/16. We report in Fig. 19 (Bottom), the maximum of the density as a function of the time iteration for different values of θ = π/16, π/8, π/4, π/2 with Type IV-PML. We again see an optimal performance with θ = π/16 but θ = π/8 or θ = π/4 also provides some good results, meaning that the PML is relatively stable with θ. In the last test, we compare in Fig. 18, the efficiency of the PML

FIG. 12. Test 2 : From top to bottom: No PML, reference and Type V-PML solutions at time t = 3.6 × 10−1 a.u..

for four different absorbing functions (Types I to IV), with respectively σ(ν) = σ0 (ν + δν )2 , σ(ν) = σ0 (ν + δν )3 , σ(ν) = −σ0 /ν and σ(ν) = σ0 /ν 2 . We see that the best results are obtained for the Type-IV PML, a similar quality being also obtained for the Types V and VI PMLs (not reported here). These tests clearly show the relevance of the combination of the TSSP method with PMLs, as we simultaneously benefit from the efficiency and accuracy of FFTs and the high absorption feature of PMLs to avoid reflections at the boundary. Test 4 : convergence. In this last test, we propose a simple convergence benchmark. A wavepacket with wavenumber k = (5, 0) is propagating in vacuum without

12

10 -2

10 -3

10 -4

10 -5

10 -6

10 -7

10 -8 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

FIG. 13. Test 2 : (Top) Solution of reference at time T = 6 × 10−2 a.u. in the domain [−18, 18] × [−18, 18]. (Bottom) `2 -norm in logscale of the 3 solutions (No PML, Reference, Type V-PML) as a function of time in D.

external electric field. We plot the initial density, and final solution with periodic solution on D = [−8, 8]2 in Fig. 20 (Top, Middle). We report the supremum of the solution for Nx = Ny = 32, 256, 320, 512, 640, 768 with the PML of Type-III, for σ0 = 10−2 , θ = π/4, fixing the time step to ∆t = 1.8×10−3 . We report the maximum of the density at final time T in logscale with the different meshes n o x, y, ln sup |d(T, x, z)| : (x, z) ∈ [−8, 8]2 . Moreover in Fig. 20 (Bottom), we observe an accurate convergence up to a certain precision. Notice that what is reported is not the comparison with a solution of reference, but the maximum of the solution over the physical domain, including the PML. In other words the smaller the maximum, the better the absorption. The saturation error comes from the limit of the PML accuracy and error in time (second-order splitting). This again shows the high accuracy of the TSSP with PML.

5.

CONCLUSION

In this paper, we have proposed a simple time-splitting pseudospectral method which allows for the numerical

FIG. 14. Test 3 : initial density in logscale (Top) and static potential (Bottom).

FIG. 15. Test 3 : logarithm of the density at time T = 0.912 (from top-left to bottom-right) for the PML with parameters σ0 = 1, σ0 = 10−1 , σ0 = 10−2 , σ0 = 10−3 , σ0 = 5, and finally for a periodic boundary condition (S = 1).

computation of IBVP for the Dirac equation with nonreflecting PML. Some numerical experiments have shown that the method preserves the accuracy of the PML with static or time-dependent external potentials. Different absorbing functions were considered and tested. The implementation strategy is developed for the TSSP approximation, but can be directly extended to other numerical schemes including implicit. We think that Fourierbased codes solving the time-dependent Dirac equation can easily be adapted to include the absorbing layers, hence drastically reducing the negative effect of periodic

13

FIG. 16. Test 3 : logarithm of the density at time T = 0.912 with null Dirichlet boundary condition. This test illustrates that artificial wave-reflecting occurs with real space methods (here a quantum lattice Boltzmann method [19]) are implemented without PML or ABC.

FIG. 17. Test 3 : logarithm of the density at time T = 0.912 for σ0 = 10−2 and θ = π/16, θ = π/8, θ = π/4 and θ = π/2.

conditions and simultaneously avoiding artificial wave reflections. We next plan to analyze mathematically the stability and convergence of the method as well as the PML accuracy for different types of absorbing functions. Acknowledgments. X. ANTOINE was partially supported by the French National Research Agency project NABUCO, grant ANR-17-CE40-0025. E. LORIN received support from NSERC through the Discovery Grant program. References.

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