comparison of split step solvers for multidimensional schrödinger problems

Computational Methods in Applied MathematicsVol. 13 (2013), No. 2, pp. 237–250c© 2013 Institute of Mathematics, NAS of BelarusDoi: 10.1515/cmam-2013-0004

Comparison of Split Step Solvers forMultidimensional Schrödinger ProblemsRaimondas Čiegis · Aleksas Mirinavičius · Mindaugas Radziunas

Abstract — This paper presents the analysis of the split step solvers for multidimen-sional Schrödinger problems. The second-order symmetrical splitting techniques areapplied. The standard operator splitting is used to split the linear diffraction and re-action/potential processes. The dimension splitting exploits the commuting propertyof one-dimensional discrete diffraction operators. Alternating Direction Implicit (ADI)and Locally One-Dimensional (LOD) algorithms are constructed and stability is inves-tigated for two- and three-dimensional problems. Compact high-order approximationsare applied to discretize diffraction operators. Results of numerical experiments arepresented and convergence of finite difference schemes is investigated.2010 Mathematical subject classification: 65N20.Keywords: Finite Difference Method, Schrödinger Problem, Split Step Method, ADIMethod, LOD Method, High Order Approximation.

1. Introduction

Many mathematical problems of nonlinear optics, laser physics, quantum mechanics aredescribed by Schrödinger problems. Usually they combine linear and nonlinear processesand are multidimensional. Therefore, the development of robust and efficient numericalalgorithms for the solution of such problems still remains a very important challenge ofcomputational mathematics.

We consider the nonlinear multidimensional Schrödinger equation

−i∂u∂t

=d∑`=1

∂2u

∂x2`

+ f(u), (1)

where (X, t) ∈ Rd×(0, T ], i =√−1 is the imaginary unity, and the dimension d ∈ 2, 3. By

assuming that the solution of the equation (1) is vanishing fast with |X| → ∞, we considerour problem on a bounded domain Q = D×(0, T ], where D = (−X1, X1)×· · ·×(−Xd, Xd) ⊂Rd, and the boundary conditions are specified by

u(X, t) = 0, X ∈ ∂D. (2)

Raimondas ČiegisVilnius Gediminas Technical University, Sauletekio al. 11, 10223 Vilnius, LithuaniaE-mail: [email protected].

Aleksas MirinavičiusVilnius Gediminas Technical University, Sauletekio al. 11, 10223 Vilnius, LithuaniaE-mail: [email protected].

Mindaugas RadziunasWeierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, GermanyE-mail: [email protected].

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238 Raimondas Čiegis, Aleksas Mirinavičius, Mindaugas Radziunas

Finally, the initial conditions are defined at t = 0:

u(X, 0) = u0(X), X ∈ D.

Function f is a complex-valued function with f(0) = 0. We assume that it satisfies theestimates

|f(v)| 6M0, |f(v)− f(w)| 6M1|v − w|

for any functions v, w in some neighborhood of the exact solution BR(u). In many applica-tions the nonlinear function f(u) is such that

Im(f(v), v) = 0, (3)

where the scalar product of two functions v, w is defined as

(v, w) =

∫· · ·∫D

v(X)w(X) dX

and w(X) denotes the complex-conjugate of w(X). A particular wide-spread case of condi-tion (3) is given by

f(u) = g(|u|2, X)u, (4)

where g is a real-valued function. Typical examples of f(u) satisfying (4) are given by acubic nonlinearity (f(u) = γ|u|2u), or by a linear potential (f(u) = V (X)u).

It is well known that the Schrödinger equation can possess a set of conservation laws.For example, our Schrödinger problem (1)–(2) with function f(u) satisfying the condition(3) preserves in time the following integral:

I1(t) :=

∫· · ·∫D

|u(X, t)|2 dX = const, t > 0. (5)

A similar conservation of a discrete approximation to Schrödinger equation is a desiredproperty of any finite difference scheme.

The main goal of the present paper is the investigation of the split step solvers for mul-tidimensional Schrödinger problems. We are interested in the Strang type second-ordersymmetrical splitting techniques. The standard operator splitting can be used to separatethe linear diffraction and reaction processes. This part of the splitting approach is well in-vestigated in many papers and books, where the first, second and high order approximationsare constructed and investigated, see [1,2,8,10,13,18,20,22]. In our paper we restricted our-selves to the symmetrical second-order splitting method. The main aim of this paper is tocompare the stability and accuracy of the Alternating Direction Implicit (ADI) and LocallyOne-Dimensional (LOD) algorithms when they are applied for two- and three-dimensionalSchrödinger problems. There are many results on splitting type algorithms used for solv-ing multidimensional parabolic problems, see [10, 16, 17]. The good stability properties ofthe Approximate Matrix Factorization (AMF) method for parabolic problems follow fromsome useful general stability results for two-stage finite difference schemes [16,17]. It is wellknown that the splitting of the diffusion operator increases the amount of the regularizationoperator. The application of the AMF method for the Maxwell equation is investigated andinteresting results are obtained in [3].

The situation is more complicated in the case of the Schrödinger problems. Thereforestability analysis and comparison of different splitting techniques for such problems is a



Comparison of Split Step Solvers for Schrödinger Problems 239

challenging task. We note that the dimension splitting of multidimensional Schrödingerproblem exploits the commuting property of one-dimensional discrete diffraction operators.

Solutions of the nonlinear Schrödinger equation subject to appropriate boundary condi-tions are known to satisfy some important conservation laws [10, 13, 18]. We are interestedin investigating the conservativity of splitting schemes when compact high-order approxima-tions of the diffraction operators are used [6].

Results of numerical experiments are presented and convergence of finite difference sche-mes is investigated.

The rest of the paper is organized as follows. In Section 2 we construct second-order sym-metric splitting schemes to separate multidimensional diffraction and reaction subproblems.Dimension splitting of the diffraction operators is done in Section 3. ADI and LOD typeapproximations are investigated for the two- and three-dimensional problems. The compacthigh-order finite difference scheme is applied to approximate diffraction operators. In Sec-tion 4 we present several computational experiments which confirm theoretical convergencerates of the constructed splitting algorithms. Finally, in Section 5 some conclusions aregiven.

2. Numerical Schemes

There are many numerical algorithms for the solution of nonlinear Schrödinger problems.They are based on various finite-difference [4, 5, 10, 19, 21], finite-element [1, 12], spectraland pseudo-spectral methods [22]. In this section we restrict our study to finite differenceschemes. We note that the main properties of discrete approximations are very similar forall mentioned methods.

We define discrete grids and grid functions Uh for the three-dimensional domain D, arestriction of this grid to the two-dimensional problem can be done trivially. The domainD := D ∪ ∂D is covered by a discrete grid which is uniform in each direction:

Ωh =Xh ∈ ω1h × ω2h × ω3h

,

where

ω`h =x`s : x`s = −X` + sh`, s = 0, . . . , J`, h` = 2X`/J`

, ` = 1, 2, 3.

The related grids on the inner part D of the spatial domain and its boundary ∂D are givenby

Ωh = ω1h × ω2h × ω3h, ∂Ωh = Ωh \ Ωh, ω`h = ω`h \ −X`, X`, ` = 1, 2, 3.

Finally,ωτ =

tn : tn = nτ, n = 0, . . . , N, Nτ = T

is a uniform time grid with the time step τ . Although the constant time step is consideredhere, the following studies can be easily extended to the case when τ varies.

We consider numerical approximations Unh = Un

jkm to the exact solution u(X, t) at thegrid points (Xh = Xjkm, t

n) ∈ Ωh×ωτ . We use the following notations for shift, difference inBrought to you by | Sterling Memorial Library

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space, and difference and averaging in time operators (the fractional time levels are defined):

T±1 Unjkm = Un

j±1,km, T±2 Unjkm = Un

j,k±1,m, T±3 Unjkm = Un

jk,m±1,

∂+xpU

nh =

T+p U

nh − Un

h

hp, ∂−xpU

nh =

Unh − T−p Un

h

hp, ∂2

xpUnh =

∂+xpU

nh − ∂−xpU

nh

hp, p = 1, 2, 3,

∂`,Lt Unh =

Un+ `

Lh − Un+ `−1

Lh

τ, A`LU

nh =

Un+ `

Lh + U

n+ `−1L

h

2, ` = 1, . . . , L.

In the case L = 1, we will use the standard notations

∂tUnh =

Un+1h − Un

h

τ, A1

1Unh =

Un+1h + Un

h

2.

2.1. Splitting of Different Processes

The finite difference scheme proposed below is based on the splitting method [2,10,13,16,17,20, 22]. The splitting and high-order discrete approximations for the Schrödinger problemsare also investigated in [1, 18].

All subsequent splitting approximations are constructed by using the following differentialsubproblems, which define the linear diffraction and nonlinear reaction problems. On eachsmall time interval [tn, tn+1] the nonlinear problem (1) is split into two subproblems (thissplitting is called Lie–Trotter [22]):

−i∂ul∂t

=d∑p=1

∂2ul∂x2

p

, X ∈ D, (6)

ul(X, t) = 0, X ∈ ∂D,

−i∂un∂t

= f(un), X ∈ D, (7)

where the initial conditions at tn and the resulting approximation of the exact solution attn+1 are given by

ul(X, tn) = δ u(X, tn) + (1− δ)un(X, tn+1),

un(X, tn) = (1− δ)u(X, tn) + δ ul(X, tn+1),

u(X, tn+1) = (1− δ)ul(X, tn+1) + δ un(X, tn+1),

respectively. Parameter δ = 1 or δ = 0 determines which of the two subproblems (6) or (7)is solved first.

For each non-negative even n and Xh ∈ Ωh we separate the diffraction and nonlinearinteraction processes by using the symmetric Strang splitting approach [10,16] (which is alsocalled the Strang–Marchuk splitting [14,15]):

−i∂1,2t Un

h =d∑p=1

∂2xpA

12 U

nh , (8)

−i∂2,2t Un

h = f(A22 U

nh ), (9)

−i∂1,2t Un+1

h = f(A12 U

n+1h ), (10)

−i∂2,2t Un+1

h =d∑p=1

∂2xpA

22 U

n+1h . (11)




Initial and boundary conditions are defined by

U0h = u0(Xh), Xh ∈ Ωh,

Unh = 0, Xh ∈ ∂Ωh, tn ∈ ωτ .

Since f(0) = 0, discrete ODEs (9)–(10) are solved only in the open domain Ωh.Even for non-commuting diffraction and reaction operators the approximation error of

the splitting scheme (9)–(11) is found to satisfy O(τ 2 + h2), where h = maxh1, . . . , hd.This estimate is obtained due to symmetry of approximations around the point tn+1.

Let H0(Ωh) denote the set of grid functions V defined on Ωh with V = 0 on ∂Ωh. Thediscrete inner product of grid functions V and W and the L2 norm on H0(Ωh) are definedas follows:

(V,W )h =

J1−1∑j=1

J2−1∑k=1

J3−1∑m=1

VjkmW jkmh1h2h3, ‖V ‖ =√

(V, V )h.

The restriction of this definition to two-dimensional spaces is obvious.Let us assume again that function f(u) satisfies condition (3). It is easy to check that the

solutions of split schemes (8)–(9) or (10)–(11) satisfy the discrete version of the conservationlaw (5),

‖Un+1‖ = ‖Un‖, ‖Un+2‖ = ‖Un+1‖ (12)

in this case. The proof follows from the standard analysis. Doing the inner product on bothsides of equation (8) with A1

2Unh , using summation by parts and taking the imaginary part

of the obtained equality, we get‖Un+ 1

2‖2 = ‖Un‖2.

Imaginary parts of both sides of (9) multiplied with A22U

nh and condition (3) give us

|Un+1h |2 =

∣∣Un+ 12

h

∣∣2 =⇒ ‖Un+1‖2 = ‖Un+ 12‖2.

After performing the same operations with equations (10) and (11) we obtain the discreteconservation law (12).

The proposed splitting template is quite flexible. For example, in the case of stiff reactionprocesses we can integrate the ODE subproblem (7) by using several smaller time stepsτs = τ/M . In this case we replace (9) by

−iM∂M+`,2Mt Un

h = f(AM+`

2M Unh

), ` = 1, . . . ,M.

If condition (3) holds, the conservation law (12) remains valid also for this modified splitstep scheme.

Once the function f(u) is given in the form (4), the solution of ODE subproblem (7)satisfies the conservation condition

|un(X, t)|2 = |un(X, tn)|2 for all t ∈ [tn, tn+1],

therefore the nonlinear ODE equation (7) can be linearized and solved explicitly. Thus, inthe case of (4), we can replace the Crank–Nicolson type scheme (9) by

Un+1h = exp

(ig(∣∣Un+ 1

2h

∣∣2)τ)Un+ 12

h , Xh ∈ Ωh.

In the same manner we can also replace (10) by the explicit solution of the correspondingODE subprocess within the time interval [tn+1, tn+2].




3. Splitting of Multidimensional Diffraction

In this section we consider splitting approximations for the multidimensional linear diffrac-tion problem given by the subprocess (6). We start the analysis from the two-dimensionalCrank–Nicolson finite-difference scheme, which can be written as

−i(I − iτ

2

2∑p=1

∂2xp

)∂tU

nh =

2∑p=1

∂2xpU

nh , Xh ∈ Ωh. (13)

3.1. ADI Scheme

In order to get the efficient solver of system (13) we perturb the matrix on the left-hand sideby using the Approximate Matrix Factorization (AMF) method [10]:

−i(I − iτ

2∂2x1

)(I − iτ

2∂2x2

)∂tU

nh =

2∑p=1

∂2xpU

nh , Xh ∈ Ωh. (14)

The influence of matrix factorization to the approximation accuracy is estimated directlysince the perturbation term can be written explicitly. We get the additional approxima-tion error term i τ

2

4∂5u

∂x21∂x22∂t

which is of order O(τ 2) for a sufficiently smooth solution of thedifferential problem.

The factorized scheme (14) can be efficiently implemented in two splitting steps [6]:

−i(I − iτ

2∂2x1

)Unh =

2∑p=1

∂2xpU

nh , Xh ∈ Ωh, (15)(

I − iτ2∂2x2

)∂tU

nh = Un

h , Xh ∈ Ωh, (16)

where each step is given by a linear tridiagonal system of equations. Thus, the AMF schemecan be seen as a particular ADI scheme.

Such a connection between the AMF and ADI methods is well understood for non-stationary diffusion-advection-reaction problems, arising in air pollution modeling, chemo-taxis models, biotechnological applications [7, 9, 10]. The good stability properties of theAMF method for parabolic problems follow from some useful general stability results fortwo-stage finite difference schemes [16,17]:

(I + τR) ∂tUnh + AUn

h = 0, U0h = φh,

where I + τR > 0 and operator R is called a regularizator. If A = A∗ > 0, i.e., operator Ais symmetric and positive definite, then a sufficient stability condition is given by

R > σ0A, σ0 =1

2− 1

τ‖A‖.

Let us assume that R = R1 + R2 > 0, operators R1, R2 commute and they are positivedefinite: R1,2 > 0. Then the following estimate is valid [16]:

(I + τR1)(I + τR2) = I + τR + τ 2R1R2 > I + τR.




Thus, for parabolic problems the AMF approximation increases the stability of the unper-turbed finite-difference scheme.

The situation becomes more complicated for the Schrödinger problem. First, we investi-gate the stability of the ADI scheme (14) with respect to the initial condition. Let λp,j beeigenvalues of operator −∂2

xp , p = 1, 2. The following estimates are valid (see [16]):

8

4X2p

6 λp,j 64

h2p

, j > 1, p = 1, 2.

According to von Neumann linear stability analysis, the numerical solution of the scheme(14) can be represented as a Fourier sum

Unjk =

J1−1∑l=1

J2−1∑m=1

cnlmY1,l(x1j)Y2,m(x2k), (17)

where Y1,l(x1)|J1−1l=1 and Y2,m(x2)|J2−1

m=1 are two sets of orthonormal eigenvectors. Substi-tuting (17) into the ADI scheme (14), we get

cn+1lm =

1− τ2

4λ1,lλ2,m − i τ2 (λ1,l + λ2,m)

1− τ2

4λ1,lλ2,m + i τ

2(λ1,l + λ2,m)

cnlm = αlmcnlm, (18)

where

αlm :=1− τ2

4λ1,lλ2,m − i τ2 (λ1,l + λ2,m)

1− τ2

4λ1,lλ2,m + i τ

2(λ1,l + λ2,m)

is the amplification factor. Since eigenvalues λp,l are real, it follows that |αlm| = 1, i.e.,|cn+1lm | = |cnlm|, and therefore

‖Un+1‖2 =

J1−1∑l=1

J2−1∑m=1

|cn+1lm |

2 =

J1−1∑l=1

J2−1∑m=1

|cnlm|2 = ‖Un‖2.

Thus, we have proved that a discrete version of the conservation law (5) is satisfied for ADIscheme (14).

The stability of the given ADI scheme with respect to the right-hand side can be alsoproved by using the spectral method. Let us consider the scheme

−i(I − iτ

2∂2x1

)(I − iτ

2∂2x2

)∂tU

nh =

2∑p=1

∂2xpU

nh +Rn

h, Xh ∈ Ωh, (19)

where the grid function Rnh can be written by means of a Fourier sum:

Rnjk =

J1−1∑l=1

J2−1∑m=1

rnlmY1,l(x1j)Y2,m(x2k).

One can easily find that equation (19) implies the following discrete equation:

cn+1lm = αlmc

nlm + τβnlmr

nlm, (20)




where the amplification factor αlm is defined above and

βnlm =i

(1 + i τ2λ1,l)(1 + i τ

2λ2,m)

=⇒ |βnlm| 6 1.

By using (20) we can write Un+1 as a sum of two functions

Un+1jk = Un+1

jk + τRn+1jk .

From the triangle inequality for the L2 norm of vectors it follows that

‖Un+1‖ 6 ‖Un+1‖+ τ‖Rn+1‖.

Taking into account the estimates of factors αlm and βnlm, we get the estimates

‖Un+1‖ 6 ‖Un‖, ‖Rn+1‖ 6 ‖Rn‖.

Therefore the stability estimate

‖Un+1‖ 6 ‖Un‖+ τ ‖Rn‖

is valid. Thus, the ADI scheme (19) is unconditionally stable when applied to the 2D linearSchrödinger equation.

3.2. LOD Scheme

Let us consider the standard LOD scheme for approximation of the 2D diffraction problem:(I − iτ

2∂2x1

)Un+ 1

2h =

(I + i

τ

2∂2x1

)Unh , Xh ∈ Ωh, (21)(

I − iτ2∂2x2

)Un+1h =

(I + i

τ

2∂2x2

)Un+ 1

2h , Xh ∈ Ωh.

Expressing the numerical solution of the LOD scheme by means of the Fourier sum (17), weget that equality (18) is valid for the Fourier coefficients. Thus, for 2D problems the LODsplit scheme is equivalent to the ADI factorization scheme (14).

3.3. High Order Approximations

In this section we consider the high-order compact finite difference schemes. Let us introducethe operator

LxpUnh =

(1 +

h2p

12∂2xp

)Unh =

1

12

(T−p U

nh + 10Un

h + T+p U

nh

), p = 1, 2,

such that for the solutions u(X, t) which are smooth enough

Lxp(∂2u

∂x2p

)= ∂2

xpu+O(h4p), p = 1, 2.

The high-order compact Numerov-type finite difference scheme [6,11] is given by

−iLx1Lx2∂tUnh = Lx2∂2

x1A1

1Unh + Lx1∂2

x2A1

1Unh , Xh ∈ Ωh. (22)

Using the Taylor expansion one can see that scheme (22) has a truncation error of O(τ 2+h4).




Remark 1. The high-order approximation will be obtained for the Strang operator splittingscheme (8)–(11) in the two-dimensional case (d = 2), if the standard central differenceapproximations of the diffraction equations (8) and (11) will be changed by the high-orderapproximations

− iLx1Lx2∂1,2t Un

h = Lx2∂2x1A1

2Unh + Lx1∂2

x2A1

2Unh ,

− iLx1Lx2∂2,2t Un

h = Lx2∂2x1A2

2Unh + Lx1∂2

x2A2

2Unh .

The ADI factorization scheme is constructed by using the same operator perturbationtechnique:

−i(Lx1 − i

τ

2∂2x1

)(Lx2 − i

τ

2∂2x2

)∂tU

nh = Lx2∂2

x1Unh + Lx1∂2

x2Unh . (23)

It can be implemented efficiently in two split steps:

−i(Lx1 − i

τ

2∂2x1

)Unh = Lx2∂2

x1Unh + Lx1∂2

x2Unh , Xh ∈ Ωh,(

Lx2 − iτ

2∂2x2

)∂tU

nh = Un

h , Xh ∈ Ωh.

For the case of non-homogeneous boundary conditions µh = µ(Xh, t) the boundary conditionsfor computing Un

h are imposed as

Unh =

(Lx2 − i

τ

2∂2x2

)∂tµ

nh, Xh ∈ −X1, X1 × ω2h.

In the case of homogeneous boundary conditions we get

Unh = 0, Xh ∈ −X1, X1 × ω2h.

Next we give information on spectral properties of the averaging operators. Let γp,j beeigenvalues of Lxp . It follows from the definition of these operators that

γp,j = 1−h2p

12λp,j =⇒ 2

36 γp,j 6 1.

The Neumann stability analysis of scheme (23) can be done similarly as for the ADIscheme (14). We restrict our study to the stability analysis with respect to the initialcondition. After simple computations we get that the Fourier coefficients of the solutionsatisfy the equation

cn+1lm =

γ1,lγ2,m − τ2

4λ1,lλ2,m − i τ2 (λ1,lγ2,m + γ1,lλ2,m)

γ1,lγ2,m − τ2

4λ1,lλ2,m + i τ

2(λ1,lγ2,m + γ1,lλ2,m)

cnlm. (24)

Then it follows that |cn+1lm | = |cnlm|, and therefore a discrete version of the conservation law

(5) is also satisfied for high-order ADI scheme (23).The high-order LOD scheme is even more simple than the ADI scheme, since only the

averaging operator corresponding to the local splitting direction should be applied:(Lx1 − i

τ

2∂2x1

)Un+ 1

2h =

(Lx1 + i

τ

2∂2x1

)Unh , Xh ∈ Ωh, (25)(

Lx2 − iτ

2∂2x2

)Un+1h =

(Lx2 + i

τ

2∂2x2

)Un+ 1

2h , Xh ∈ Ωh.

Again, if we express the numerical solution of the high-order LOD scheme by means ofthe Fourier sum, then we get that equality (24) is valid for the Fourier coefficients. Thus,for 2D problems the high-order LOD split scheme (25) is equivalent to the ADI factorizationscheme (23).




3.4. Splitting Schemes for the 3D Problem

In this section we consider generalizations of ADI and LOD schemes for the three-dimensionaldiffraction problem. By using the AMF method we write a factorized scheme:

− i(I−iτ

2∂2x1

)(I−iτ

2∂2x2

)(I−iτ

2∂2x3

)∂tU

nh =

3∑p=1

∂2xpU

nh , Xh ∈ Ωh. (26)

It can be implemented as a three-step ADI scheme similarly to (15)–(16).Following the von Neumann method for linear stability analysis we write the numerical

solution to scheme (26) by means of the Fourier sum

Unjkm =

J1−1∑l=1

J2−1∑r=1

J3−1∑s=1

cnlrsYl(x1j)Yr(x2k)Ys(x3m). (27)

Substituting (27) into the ADI scheme (26), we get cn+1lrs = αlrsc

nlrs, where amplification factor

αlrs is given by

αlrs =βlrs − i τ2 (λ1,l + λ2,r + λ3,s + τ2

4λ1,lλ2,rλ3,s)

βlrs + i τ2(λ1,l + λ2,r + λ3,s − τ2

4λ1,lλ2,rλ3,s)

,

βlrs = 1− τ 2

4(λ1,lλ2,r + λ1,lλ3,s + λ2,rλ3,s).

Since eigenvalues λ are positive, we have |αlrs| > 1 and, therefore, the ADI scheme is unstablefor the 3D diffraction problem.

Remark 2. We note that for three-dimensional parabolic problems such ADI scheme isunconditionally stable. This result follows from the general stability results given above.

Next we consider the following LOD scheme:(I − iτ

2∂2x`

)Un+ `

3h =

(I + i

τ

2∂2x`

)Un+

(`−1)3

h , ` = 1, 2, 3, Xh ∈ Ωh. (28)

The amplification factor αlrs of this scheme reads

αlrs =(1− i τ

2λ1,l)(1− i τ2λ2,r)(1− i τ2λ3,s)

(1 + i τ2λ1,l)(1 + i τ

2λ2,r)(1 + i τ

2λ3,s)

.

It is easy to see that |αlrs| = 1, and, therefore, a discrete version of the conservation law (5)is satisfied for the three-dimensional LOD scheme (28).

By comparing amplification factors αlrs for three-dimensional ADI and LOD schemeswe can find that the LOD splitting introduces the additional perturbation term in the ADIscheme:

−i(I − iτ

2∂2x1

)(I − iτ

2∂2x2

)(I − iτ

2∂2x3

)∂tU

nh =

3∑p=1

∂2xpU

nh +

τ 2

4∂2x1∂2x2∂2x3Unh , Xh ∈ Ωh.

It follows from this equality that for sufficiently smooth solutions the approximation errorof the LOD splitting scheme is of order O(τ 2 + h2).




Finally, we formulate a general splitting scheme approximating the nonlinear d-dimen-sional Schrödinger equation (1). This scheme includes the symmetric Strang separation ofdifferent processes and the LOD method for treatment of the multidimensional diffractionoperator: (

T` − iτ

2∂2x`

)Un+ `

d+1

h =(T` + i

τ

2∂2x`

)Un+

(`−1)d+1

h , ` = 1, . . . , d,

−i∂d+1,d+1t Un

h = f(Ad+1d+1 U

nh

), −i∂1,d+1

t Un+1h = f

(A1d+1 U

n+1h

),(

T` − iτ

2∂2x`

)Un+1+ `

d+1

h =(T` + i

τ

2∂2x`

)Un+1+

(`−1)d+1

h , ` = 1, . . . , d,

T` := θI+(1−θ)Lx` , θ = 0, 1.

(29)

Here θ = 1 and θ = 0 define the standard central differences approximation and the high-order approximation of the diffraction operator, respectively.

4. Computational Experiments

In this section we present numerical results for two examples of two-dimensional linearSchrödinger problems. The first example represents the case of a non-constant potentialand of non-commuting diffraction and reaction operators. It is used to investigate the con-vergence order of the symmetrical Strang scheme which has a formal consistency order oftwo. In the second example the potential is constant and, therefore, the operators are com-muting. This example is used to investigate the accuracy of the high-order approximationof the diffraction operator. Splitting scheme (29) is used in all numerical experiments.

Example 1. Consider the linear Schrödinger equation [6]

−i∂u∂t

=∂2u

∂x2+∂2u

∂y2+(3− 2 tanh2(x)− 2 tanh2(y)

)u

in the domain [−20, 20]× [−20, 20]. The problem is solved till T = 1. The exact solution isgiven as

u(x, y, t) =i exp(it)

cosh(x) cosh(y),

what allows us to use the homogeneous boundary conditions in the given domain. The initialconditions are obtained from the exact solution.

The goal is to investigate the temporal convergence rate of the splitting scheme (29).Computations have been performed with θ = 1, different time steps τ = 1/N , and verysmall spatial grid steps. Table 1 presents the errors

‖Z‖∞ = maxXjk∈Ωh

|UNjk − u(Xjk, T )|.

The results of the computational experiments are in good agreement with the theoreticalconvergence estimate O(τ 2) for the symmetrical Strang splitting and the ADI dimensionalsplitting algorithm.




N = 10 N = 20 N = 40 N = 80 N = 160

‖Z‖∞ 0.022426 0.005785 0.001458 0.000365 0.000092

Table 1. Convergence analysis of the splitting scheme (29) with respect to the temporal integration. Errorsof the discrete solution are presented at T = 1. A sequence of step-sizes τ = T/N is used.

N = 10, J = 512 N = 20, J = 1024 N = 40, J = 2048

Scheme (21) 0.003614 0.000911 0.000228

N = 10, J = 100 N = 20, J = 141 N = 40, J = 200

Scheme (25) 0.003780 0.000949 0.000236

Table 2. Comparison of the standard (28) and high order (25) approximations of the diffraction operatorin the splitting scheme (29). Errors of the discrete solution are presented at T = 0.1 for a a sequence ofstep-sizes τ = T/N , h = 40/J .

Example 2. Consider the linear Schrödinger equation [6]

−i∂u∂t

=∂2u

∂x2+∂2u

∂y2− u

in the domain [−10, 10]× [−10, 10]. The problem is solved till T = 0.1. The exact solutiondescribes the movement of the transient Gaussian soliton

u(x, y, t) =ie−it

i− 4texp[−i((x−1)2+(y−1)2 + ik(x− 1) + ik2t

)/(i−4t)

],

where (1, 1) is the initial center of the soliton, and k = 2.5 is the wave number. Like in theprevious example, we use the homogeneous boundary conditions and the initial conditionscomputed from the exact solution.

We have compared the accuracy of the standard central differences approximation of thediffraction operator (21) and the high-order approximation scheme (25). The test problem issolved by using the following relations between mesh steps: τ = Ch for the (21) scheme andτ = Ch2 for the high-order scheme. The results of computations are presented in Table 2.

The results of the computational experiments have confirmed the theoretical convergenceestimate O(τ 2 + h2) for the central differences approximation scheme (28) and O(τ 2 + h4)for the high order (25) approximation.

5. Conclusions

In this paper we have studied the AMF and LOD splitting schemes for two- and three-dimen-sional nonlinear Schrödinger problems. We use Strang’s symmetrical splitting technique as abasic method. The diffraction operator is approximated by the standard second order centraldifferences scheme and by compact high-order scheme. It is proved that for two-dimensionalproblems both AMF and LOD splitting schemes are stable and preserve the discrete L2 normof the solution. For tree-dimensional problems only the LOD scheme is stable. Results ofcomputational experiments are presented, which confirm theoretical convergence rates ofthe constructed splitting algorithms. It would be interesting to compare the efficiency of theconstructed solvers with the efficiency of solvers based on multigrid and FFT techniques.




Acknowledgments

The authors would like to thank the referees for their constructive criticism which helped toimprove the clarity and quality of this note. The work of M. Radziunas was supported byDFG Research Center Matheon “Mathematics for key technologies: Modelling, simulationand optimization of the real world processes”.

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comparison of split step solvers for multidimensional schrödinger problems

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