Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular

30 Pages·2012·0.64 MB·English

MATHEMATICSOFCOMPUTATION
Volume82,Number281,January2013,Pages99–128
S0025-5718(2012)02617-2
ArticleelectronicallypublishedonJune20,2012
OPTIMAL ERROR ESTIMATES OF FINITE DIFFERENCE
METHODS FOR THE GROSS-PITAEVSKII EQUATION
WITH ANGULAR MOMENTUM ROTATION
WEIZHUBAOANDYONGYONGCAI
Abstract. WeanalyzefinitedifferencemethodsfortheGross-Pitaevskiiequa-
tion with an angular momentum rotation term in two and three dimensions
andobtaintheoptimalconvergencerate,fortheconservativeCrank-Nicolson
finite difference (CNFD) method and semi-implicit finite difference (SIFD)
method,attheorderofO(h2+τ2)inthel2-normanddiscreteH1-normwith
time step τ and mesh size h. Besides the standard techniques of the energy
method, the key technique in the analysis for the SIFD method is to use the
mathematicalinduction,andresp.,fortheCNFDmethodistoobtainapriori
boundofthenumericalsolutioninthel∞-normbyusingtheinverseinequality
andthel2-normerrorestimate. Inaddition,fortheSIFDmethod,wealsode-
riveerrorboundsontheerrorsbetweenthemassandenergyinthediscretized
levelandtheircorrespondingcontinuouscounterparts,respectively,whichare
at the same order of the convergence rate as that of the numerical solution
itself. Finally,numericalresultsarereportedtoconfirmourerrorestimatesof
thenumericalmethods.
1. Introduction
In this paper, we analyze different finite difference approximations of the Gross-
Pitaevskii equation (GPE) with an angular momentum rotation term in d-
dimensions (d = 2,3) for modeling a rotating Bose-Einstein condensate (BEC)
[35, 12]:
(cid:2) (cid:3)
1
i∂ ψ(x,t)= − ∇2+V(x)−ΩL +β|ψ(x,t)|2 ψ(x,t),
(1.1) t 2 z
x∈U ⊂Rd, t>0,
with the homogeneous Dirichlet boundary condition
(1.2) ψ(x,t)=0, x∈Γ=∂U, t≥0,
and initial condition
(1.3) ψ(x,0)=ψ (x), x∈U.
0
Heretistime,x=(x,y)intwodimensions(2D),i.e.,d=2,andresp.,x=(x,y,z)
in three dimensions (3D), i.e., d=3, are the cartesian coordinates, U is a bounded
computational domain, ψ := ψ(x,t) is the complex-valued wave function, Ω is a
dimensionless constant corresponding to the angular speed of the laser beam in
ReceivedbytheeditorMarch29,2010and,inrevisedform,March26,2011.
2010MathematicsSubjectClassification. Primary35Q55,65M06,65M12,65M22,81-08.
Key words and phrases. Gross-Pitaevskiiequation,angularmomentumrotation,finitediffer-
encemethod,semi-implicitscheme,conservativeCrank-Nicolsonscheme.
(cid:3)c2012AmericanMathematicalSociety
Reverts to public domain 28 years from publication
99
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100 WEIZHUBAOANDYONGYONGCAI
experiments, β is a dimensionless constant characterizing the interaction (positive
for repulsive interaction and negative for attractive interaction) between particles
in the rotating BEC. V(x) is a real-valued function corresponding to the external
trappotentialanditischosenasaharmonicpotential,i.e.,aquadraticpolynomial,
in most experiments. L is the z-component of the angular momentum defined as
z
(1.4) L =−i(x∂ −y∂ )=−i∂ ,
z y x θ
where(r,θ)and(r,θ,z)arethepolarcoordinatesin2Dandcylindricalcoordinates
in 3D, respectively. In fact, since the first experimental realization of BEC in 1995
[5, 18] and the observation of quantized vortices in rotating BEC [1, 14, 32] which
is related to superfluidity, theoretical studies of BEC and quantized vortices based
ontheaboveGPEhavestimulatedgreatresearchinterestsinquantumphysicsand
applied and computational mathematics communities.
Extensive mathematical and numerical studies have been carried out for the
aboveGPE(1.1)intheliterature. Alongthemathematicalfront,forthederivation,
well-posedness and dynamical properties of the GPE (1.1) with (i.e., Ω (cid:6)= 0) and
without(i.e.,Ω=0)anangularmomentumrotationterm,wereferto[15,23,24,29]
and the references therein. In fact, it is easy to show that the GPE (1.1) conserves
the total mass
(cid:4)
(1.5) N(ψ(·,t)):= |ψ(x,t)|2dx≡N(ψ(·,0))=N(ψ ), t≥0,
0
U
and the energy
(1.6) (cid:4) (cid:2) (cid:3)
1 1
E(ψ(·,t)):= |∇ψ|2+V(x)|ψ|2+ β|ψ|4−Ωψ¯L ψ dx≡E(ψ ), t≥0,
2 2 z 0
U
where f¯ denotes the conjugate of f. Along the numerical front, different effi-
cient and accurate numerical methods including the time-splitting pseudospectral
method [7, 25, 36, 37], finite difference method [2, 3], and Runge-Kutta or Crank-
Nicolsonpseudospectral method[14, 20] have beendevelopedfor theGPE without
and with [6, 9, 11] the angular momentum rotation term. Of course, each method
hasitsadvantagesanddisadvantages. Fornumericalcomparisonsbetweendifferent
numericalmethodsforGPEwithoutangularmomentumrotation,orinamoregen-
eral case, for the nonlinear Schro¨dinger (NLS) equation, we refer to [8, 17, 31, 39]
and references therein.
ErrorestimatesfordifferentnumericalmethodsofNLS,e.g. theGPE(1.1)with-
outtheangularmomentumrotation(Ω=0)and/ord=1,havebeenestablishedin
the literatures. For the analysis of splitting error of the time-splitting or split-step
method for NLS, we refer to [13, 19, 30, 33, 38] and the references therein. For the
errorestimatesoftheimplicitRunge-KuttafiniteelementmethodforNLS,werefer
to [4, 34]. Error bounds of conservative Crank-Nicolson finite difference (CNFD)
method for NLS in 1D was established in [16, 21, 22, 41]. In fact, their proofs for
CNFD rely strongly on the conservative property of the method and the discrete
version of the Sobolev inequality in 1D
(cid:8)f(cid:8)2 ≤(cid:8)∇f(cid:8) ·(cid:8)f(cid:8) , ∀f ∈H1(U) with U ⊂R,
L∞ L2 L2 0
which immediately imply an a priori uniform bound for (cid:8)f(cid:8)L∞. However, the ex-
tension of the discrete version of the above Sobolev inequality is no longer valid
in 2D and 3D. Thus the techniques used in [16, 21] for obtaining error bounds
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OPTIMAL ERROR ESTIMATES OF FINITE DIFFERENCE METHODS 101
of CNFD for NLS only work for conservative schemes in 1D and they cannot be
extended to either high dimensions or nonconservative finite difference schemes.
To our knowledge, no error estimates are available in the literature of finite differ-
ence methods for NLS either in high dimensions or for a non-conservative scheme.
However, the GPE with the angular momentum rotation is either in 2D or 3D
[6, 9, 11, 35]. The main aim of this paper is to use different techniques to estab-
lish optimal error bounds of CNFD and the semi-implicit finite difference (SIFD)
method for the GPE (1.1) with the angular momentum rotation in 2D and 3D.
Based on our results, both CNFD and SIFD have the same second-order conver-
gence rate in space and time. In our analysis, besides the standard techniques of
the energy method, for SIFD, we adopt the mathematical induction; for CNFD,
we first derive the l2-norm error estimate and then obtain an a priori bound of the
numerical solution in the l∞-norm by using the inverse inequality.
The paper is organized as follows. In section 2, we present SIFD and CNFD for
the GPE with the angular momentum rotation and state our main error estimate
results. Insection3,optimalerrorboundsofSIFDforGPEareestablishedbyusing
the energy method and the mathematical induction; and optimal error estimates
of CNFD for GPE is built in section 4. In section 5, extensions of our results to
other cases are discussed. In section 6, numerical results are reported to confirm
our error estimates. Finally, some conclusions are drawn in section 7. Throughout
the paper, we adopt the standard Sobolev spaces and their corresponding norms,
let C denote a generic constant which is independent of mesh size h and time step
τ, and use the notation p (cid:2) q to represent that there exists a generic constant C
which is independent of time step τ and mesh size h such that |p|≤Cq.
2. Finite difference methods and main results
In this section, we introduce SIFD and CNFD methods for the GPE (1.1) in 2D
onarectangleU =[a,b]×[c,d],andresp.,in3DonacubeU =[a,b]×[c,d]×[e,f],
and state our main error estimate results.
2.1. Numerical methods. For the simplicity of notation, we only present the
methods in 2D, i.e., d = 2 and U = [a,b]×[c,d] in (1.1). Extensions to 3D are
straightforward, and the error estimates in l2-norm and discrete H1-norm are the
same in 2D and 3D. Choose time step τ := Δt and denote time steps as t :=nτ
n
forn=0,1,2,...; choose meshsizesΔx:= b−a andΔy := d−c withM andK two
M K
positive integers and denote h := h = max{Δx, Δy}, h := min{Δx,Δy}
max min
and grid points as
x :=a+jΔx, j =0,1,...,M; y :=c+kΔy, k =0,1,...,K.
j k
Define the index sets
T ={(j,k) | j =1,2,...,M −1, k =1,2,...,K−1},
M
T0 ={(j,k) | j =0,1,2...,M, k =0,1,2...,K}.
M
Let ψn be the numerical approximation of ψ(x ,y ,t ) for (j,k) ∈T0 and n≥0
jk j k n M
anddenoteψn ∈C(M+1)×(K+1) bethenumericalsolutionattimet=t . Introduce
n
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102 WEIZHUBAOANDYONGYONGCAI
the following finite difference operators:
1 1
δ+ψn = (ψn −ψn ), δ+ψn = (ψn −ψn ),
x jk Δx j+1k jk y jk Δy jk+1 jk
1 1
δ+ψn = (ψn+1−ψn ), δ−ψn = (ψn −ψn ),
t jk τ jk jk x jk Δx jk j−1k
1 1
δ−ψn = (ψn −ψn ), δ−ψn = (ψn −ψn−1),
y jk Δy jk jk−1 t jk τ jk jk
ψn −ψn ψn −ψn
δ ψn = j+1k j−1k, δ ψn = jk+1 jk−1,
x jk 2Δx y jk 2Δy
ψn+1−ψn−1 ψn −2ψn +ψn
δ ψn = jk jk , δ2ψn = j+1k jk j−1k,
t jk 2τ x jk (Δx)2
ψn −2ψn +ψn
δ2ψn = jk+1 jk jk−1, (j,k)∈T ,
y jk (Δy)2 M
δ+ψn =(δ+ψn ,δ+ψn ), δ2ψn =δ2ψn +δ2ψn ,
∇ jk x jk y jk ∇ jk x jk y jk
Lhψn =−i(x δ ψn −y δ ψn ).
z jk j y jk k x jk
Thenthe conservativeCrank-Nicolson finite difference(CNFD)discretizationof
the GPE (1.1) reads
(2.1) (cid:2) (cid:3)
1 β
iδ+ψn = − δ2 +V −ΩLh+ (|ψn+1|2+|ψn |2) ψn+1/2, (j,k)∈T , n≥0,
t jk 2 ∇ jk z 2 jk jk jk M
where
(cid:5) (cid:6)
1
V =V(x ,y ), ψn+1/2 = ψn+1+ψn , (j,k)∈T0, n=0,1,2,....
jk j k jk 2 jk jk M
The boundary condition (1.2) is discretized as
(2.2) ψn =ψn =0, ψn =ψn =0, (j,k)∈T0, n=0,1,... ,
0k Mk j0 jK M
and the initial condition (1.3) is discretizaed as
(2.3) ψ0 =ψ (x ,y ), (j,k)∈T0.
jk 0 j k M
As proven in section 4, the above CNFD method conserves the mass and energy in
the discretized level. However, it is a fully implicit method, i.e., at each time step,
a fully nonlinear system must be solved, which may be very expensive, especially
in 2D and 3D. In fact, if the fully nonlinear system is not solved numerically to
extremely high accuracy, e.g., at machine accuracy, then the mass and energy of
the numerical solution obtained in practical computation are no longer conserved.
This motivates us also consider the following discretization for the GPE.
The semi-implicit finite difference (SIFD) discretization for the GPE (1.1) is
to use Crank-Nicolson/leap-frog schemes for discretizing linear/nonlinear terms,
respectively, as
(2.4)
(cid:2) (cid:3)
1 ψn+1+ψn−1
iδ ψn = − δ2 +V −ΩLh jk jk +β|ψn |2ψn , (j,k)∈T , n≥1.
t jk 2 ∇ jk z 2 jk jk M
Again, the boundary condition (1.2) and initial condition (1.3) are discretized in
(2.2) and (2.3), respectively. In addition, the first step can be computed by any
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OPTIMAL ERROR ESTIMATES OF FINITE DIFFERENCE METHODS 103
explicitsecondorhigherordertimeintegrator,e.g.,thesecond-ordermodifiedEuler
method, as
(cid:2)(cid:7) (cid:8) (cid:3)
1
ψ1 =ψ0 −iτ − δ2 +V −ΩLh ψ(1)+β|ψ(1)|2ψ(1) , (j,k)∈T ,
jk jk 2 ∇ jk z jk jk jk M
(2.5) (cid:2)(cid:7) (cid:8) (cid:3)
τ 1
ψ(1) =ψ0 −i − δ2 +V −ΩLh ψ0 +β|ψ0 |2ψ0 .
jk jk 2 2 ∇ jk z jk jk jk
ForthisSIFDmethod,ateachtimestep,onlyalinearsystemistobesolved,which
is much less expensive than that of the CNFD method in practical computation.
2.2. Main error estimate results. Before we state our main error estimate re-
sults, we denote the space
(cid:9) (cid:10)
X = u=(u ) | u =u =u =u =0, (j,k)∈T0
M jk (j,k)∈T0 0k Mk j0 jK M
M
⊂C(M+1)×(K+1),
and define norms and inner product over X as
M
M(cid:11)−1K(cid:11)−1
(2.6) (cid:8)u(cid:8)2 =ΔxΔy |u |2,
2 jk
j=0 k=0
M(cid:11)−1K(cid:11)−1(cid:5)(cid:12) (cid:12) (cid:12) (cid:12) (cid:6)
(cid:8)δ+u(cid:8)2 =ΔxΔy (cid:12)δ+u (cid:12)2+(cid:12)δ+u (cid:12)2 ,
∇ 2 x jk y jk
j=0 k=0
(2.7) (cid:8)u(cid:8)∞ = sup |ujk|,
0≤j≤M−1,0≤k≤K−1
M(cid:11)−1K(cid:11)−1
(cid:8)u(cid:8)p =ΔxΔy |u |p, 0<p<∞,
p jk
j=0 k=0
M(cid:11)−1K(cid:11)−1(cid:13) (cid:14)
1
(2.8) E(u)= (cid:8)δ+u(cid:8)2+ΔxΔy V |u |2−Ωu¯ Lhu , ∀u∈X ,
2 ∇ 2 jk jk jk z jk M
j=1 k=1
M(cid:11)−1K(cid:11)−1(cid:13) (cid:14)
1 β
(2.9) E (u)= (cid:8)δ+u(cid:8)2+ (cid:8)u(cid:8)4+ΔxΔy V |u |2−Ωu¯ Lhu ,
h 2 ∇ 2 2 4 jk jk jk z jk
j=1 k=1
M(cid:11)−1K(cid:11)−1
(2.10) (u,v)=ΔxΔy u v¯ ,
jk jk
j=0 k=0
M(cid:11)−1K(cid:11)−1
(cid:12)u,v(cid:13)=ΔxΔy u v¯ , ∀u,v ∈X .
jk jk M
j=1 k=1
We also make the following assumptions:
(A)AssumptiononthetrappingpotentialV(x)androtationspeedΩ, i.e., there
exists a constant γ >0 such that
1
V(x)∈C1(U), V(x)≥ γ2(x2+y2), ∀x∈U, |Ω|<γ;
2
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104 WEIZHUBAOANDYONGYONGCAI
and assumption on the exact solution ψ, i.e., let 0 < T < T with T be the
max max
maximal existing time of the solution [15, 23]:
ψ ∈C3([0,T];W1,∞(U))∩C2([0,T];
(B)
W3,∞(U))∩C0([0,T];W5,∞(U)∩H1(U)).
0
Define the “error” function en ∈X as
M
(2.11) en =ψ(x ,y ,t )−ψn , (j,k)∈T0, n≥0.
jk j k n jk M
Then for the SIFD method, we have
Theorem 2.1 (Convergence of SIFD). Assume h (cid:2) h and τ (cid:2) h, under as-
min
sumptions (A) and (B), there exist h >0 and 0<τ < 1 sufficiently small, when
0 0 4
0 < h ≤ h and 0 < τ ≤ τ , we have the following optimal error estimate for the
0 0
SIFD method (2.4) with (2.2), (2.3) and (2.5)
T
(2.12) (cid:8)en(cid:8) (cid:2)h2+τ2, (cid:8)δ+en(cid:8) (cid:2)h3/2+τ3/2, 0≤n≤ .
2 ∇ 2 τ
In addition, if either Ω = 0 and V(x) = 0 or ψ ∈ C0([0,T];H2(U)), we have the
0
optimal error estimates:
T
(2.13) (cid:8)en(cid:8) +(cid:8)δ+en(cid:8) (cid:2)h2+τ2, 0≤n≤ .
2 ∇ 2 τ
Similarly, for the CNFD method, we have
Theorem 2.2 (Convergence(cid:5)ofCNF(cid:6)D). Supposeh(cid:2)hmin, τ (cid:2)h and either β ≥0
or β < 0 with (cid:8)ψ0(cid:8)2 < 1 1− Ω2 , under assumption (A), there exists h > 0
2 |β| γ2 0
sufficiently small, when 0 < h ≤ h , the discretization (2.1) with (2.2) and (2.3)
0
admits a unique solution ψn (0 ≤ n ≤ T). Furthermore, under assumption (B),
τ
there exist h >0 and τ >0 sufficiently small, when 0<h ≤h and 0<τ ≤τ ,
0 0 0 0
we have the following error estimate:
T
(2.14) (cid:8)en(cid:8) (cid:2)h2+τ2, (cid:8)δ+en(cid:8) (cid:2)h3/2+τ3/2, 0≤n≤ .
2 ∇ 2 τ
In addition, if either Ω = 0 and V(x) = 0 or ψ ∈ C0([0,T];H2(U)), we have the
0
optimal error estimates:
T
(2.15) (cid:8)en(cid:8) +(cid:8)δ+en(cid:8) (cid:2)h2+τ2, 0≤n≤ .
2 ∇ 2 τ
3. Error estimates for the SIFD method
In this section, we establish optimal error estimates for the SIFD method (2.4)
with (2.2), (2.3) and (2.5) in l2-norm, discrete H1-norm and l∞-norm. Let ψn ∈
X bethenumericalsolutionoftheSIFDmethodanden ∈X theerrorfunction.
M M
From (2.8) and (2.10), we have
Lemma 3.1. The following equalities hold:
(cid:15) (cid:16) (cid:17) (cid:18)
(3.1) (cid:12)δ u,v(cid:13)=−(cid:12)u,δ v(cid:13), δ2u,v =− δ+u,δ+v ,
x x (cid:15) x (cid:16) (cid:17) x x (cid:18)
(3.2) (cid:12)δ u,v(cid:13)=−(cid:12)u,δ v(cid:13), δ2u,v =− δ+u,δ+v , ∀u,v ∈X ,
y y y y y M
(3.3) (cid:8)u(cid:8)2 (cid:2)(cid:8)δ+u(cid:8)2, (cid:8)u(cid:8)4 ≤(cid:8)u(cid:8)2·(cid:8)δ+u(cid:8)2, ∀u∈X .
2 ∇ 2 4 2 ∇ 2 M
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OPTIMAL ERROR ESTIMATES OF FINITE DIFFERENCE METHODS 105
In addition, under assumption (A), we have
(cid:7) (cid:8)
1 Ω2
(3.4) 1− (cid:8)δ+u(cid:8)2 ≤E(u)(cid:2)(cid:8)δ+u(cid:8)2+(cid:8)u(cid:8)2 (cid:2)(cid:8)δ+u(cid:8)2, ∀u∈X .
2 γ2 ∇ 2 ∇ 2 2 ∇ 2 M
Proof. The equality (3.1) follows from (2.10) by using summation by parts as
(cid:12)δ u,v(cid:13) = ΔxΔyM(cid:11)−1K(cid:11)−1uj+1k−uj−1k v¯
x 2Δx jk
j=1 k=1
= ΔxΔyM(cid:11)−1K(cid:11)−1u v¯j−1k−v¯j+1k =−(cid:12)u,δ v(cid:13),
jk 2Δx x
j=1 k=1
(cid:15)δ2u,v(cid:16) = ΔxΔyM(cid:11)−1K(cid:11)−1uj+1k−2ujk+uj−1k v¯
x (Δx)2 jk
j=1 k=1
M(cid:11)−1K(cid:11)−1u −u v¯ −v¯
= ΔxΔy j+1k jk j,k j+1k
Δx Δx
(cid:17) j=0 (cid:18)k=0
= − δ+u,δ+v , ∀u,v ∈X .
x x M
Similarly, we can get (3.2). For u∈X , we have
M
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)(cid:11)j−1(cid:19) (cid:20)(cid:12) (cid:12)(cid:11)j−1 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)(u )2(cid:12) = (cid:12) (u )2−(u )2 (cid:12)=Δx(cid:12) [u +u ]δ+u (cid:12)
jk (cid:12) l+1k lk (cid:12) (cid:12) l+1k lk x lk(cid:12)
l=0 l=0
(cid:11)j−1 (cid:12) (cid:12)
≤ Δx |u +u |·(cid:12)δ+u (cid:12)
l+1k lk x lk
l=0(cid:21) (cid:21)
(cid:22) (cid:22)
√ (cid:22)M(cid:11)−1 (cid:22)M(cid:11)−1
(cid:23) (cid:23)
(3.5) ≤ 2Δx |δ+u |2 |u |2, (j,k)∈T .
x lk lk M
l=0 l=0
Similarly, we have
(cid:21) (cid:21)
(cid:22) (cid:22)
(cid:12)(cid:12) (cid:12)(cid:12) √ (cid:22)(cid:23)K(cid:11)−1 (cid:22)(cid:23)K(cid:11)−1
(3.6) (cid:12)(u )2(cid:12)≤ 2Δy |δ+u |2 |u |2, (j,k)∈T .
jk y jm jm M
m=0 m=0
Combining (3.5) and (3.6), using the Cauchy inequality, we get
M(cid:2)−1K(cid:2)−1 M(cid:2)−1K(cid:2)−1
(cid:2)u(cid:2)4 =ΔxΔy |u |4 =ΔxΔy |u |2·|u |2
4 jk jk jk
j=0 k=0 ⎛(cid:5) j=0(cid:5)k=0 (cid:5) (cid:5) ⎞
(cid:6) (cid:6) (cid:6) (cid:6)
M(cid:2)−1K(cid:2)−1 (cid:6)M(cid:2)−1 (cid:6)M(cid:2)−1 (cid:6)K(cid:2)−1 (cid:6)K(cid:2)−1
≤2(ΔxΔy)2 ⎝(cid:7) |δ+u |2(cid:7) |u |2(cid:7) |δ+u |2(cid:7) |u |2⎠
x lk lk y jm jm
j=0 k=0 l=0 l=0 m=0 m=0
⎛(cid:5) (cid:5) ⎞ ⎛(cid:5) (cid:5) ⎞
(cid:6) (cid:6) (cid:6) (cid:6)
K(cid:2)−1 (cid:6)M(cid:2)−1 (cid:6)M(cid:2)−1 M(cid:2)−1 (cid:6)K(cid:2)−1 (cid:6)K(cid:2)−1
=2(ΔxΔy)2 ⎝(cid:7) |δ+u |2(cid:7) |u |2⎠ ⎝(cid:7) |δ+u |2(cid:7) |u |2⎠
x lk lk y jm jm
k=0 l=0 l=0 j=0 m=0 m=0
(cid:5) (cid:5) (cid:5) (cid:5)
(cid:6) (cid:6) (cid:6) (cid:6)
(cid:6)K(cid:2)−1M(cid:2)−1 (cid:6)K(cid:2)−1M(cid:2)−1 (cid:6)M(cid:2)−1K(cid:2)−1 (cid:6)M(cid:2)−1K(cid:2)−1
≤2(ΔxΔy)2(cid:7) |δ+u |2(cid:7) |u |2(cid:7) |δ+u |2(cid:7) |u |2
x lk lk y jm jm
k=0 l=0 k=0 l=0 j=0 m=0 j=0 m=0
≤(cid:2)δ+u(cid:2)2·(cid:2)u(cid:2)2, u∈X .
∇ 2 2 M
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106 WEIZHUBAOANDYONGYONGCAI
The first inequality in (3.3) can be proved in a similar way. From (2.8) and
summation by parts, we get
M(cid:11)−1K(cid:11)−1 M(cid:11)−1K(cid:11)−1
u¯ Lhu = −i u¯ (x δ u −y δ u )
jk z jk jk j y jk k x jk
j=1 k=1 j=1 k=1
M(cid:11)−1K(cid:11)−1
= −i u (x δ u¯ −y δ u¯ )
jk j y jk k x jk
j=1 k=1
M(cid:11)−1K(cid:11)−1
(3.7) = u L¯hu¯ ∈R, ∀u∈X ,
jk z jk M
j=1 k=1
which immediately implies that E(u) ∈ R for all u ∈ X . In addition, using
M
the Cauchy inequality and triangle inequality, noticing assumption (A), we get for
u∈X ,
M
M(cid:11)−1K(cid:11)−1
−Ω u¯ Lhu
jk z jk
j=1 k=1
M(cid:11)−1K(cid:11)−1 (cid:13) (cid:17) (cid:18) (cid:17) (cid:18)(cid:14)
Ω
(3.8) = 2 iu¯jk xj δy+ujk+δy+uj,k−1 −yk δx+ujk+δx+uj−1,k
j=1 k=1
(cid:2) (cid:3)
M(cid:11)−1K(cid:11)−1 Ω2 (cid:17) (cid:18)
≥− V |u |2+ |δ+u |2+|δ+u |2 .
jk jk 2γ2 x jk y jk
j=0 k=0
Plugging (3.8) into (2.8) and noticing (2.6), we get (3.4) immediately. (cid:3)
From now on, without loss of generality, we assume that Δx = Δy = h. From
(3.4) in Lemma 3.1, we have
Lemma 3.2 (Solvability of the difference equations). Under assumption (A), for
any given initial data ψ0 ∈ X , there exists a unique solution ψn ∈ X of (2.5)
M M
for n=1 and (2.4) for n>1.
Proof. Theassertionforn=1isobviouslytrue. InSIFD(2.5),forgivenψn−1,ψn ∈
X (n ≥ 1), we first prove the uniqueness. Suppose there exist two solutions
M
ψ(1),ψ(2) ∈X satisfying the SIFD scheme (2.4), i.e., for (j,k)∈T ,
M M
(cid:2) (cid:3)
ψ(1)−ψn−1 1 ψ(1)+ψn−1
(3.9) i jk jk = − δ2 +V −ΩLh jk jk +β|ψn |2ψn ,
2τ 2 ∇ jk z 2 jk jk
(cid:2) (cid:3)
ψ(2)−ψn−1 1 ψ(2)+ψn−1
(3.10) i jk jk = − δ2 +V −ΩLh jk jk +β|ψn |2ψn .
2τ 2 ∇ jk z 2 jk jk
Denote u=ψ(1)−ψ(2) ∈X and subtract (3.10) from (3.9), we have
(cid:2) M (cid:3)
u 1
(3.11) i jk = − δ2 +V −ΩLh u , (j,k)∈T .
τ 2 ∇ jk z jk M
Multiplying both sides of (3.11) by u¯ and summing together for (j,k) ∈ T ,
jk M
using the summation by parts formula and taking imaginary parts, using (3.4)
from Lemma 3.1, we obtain (cid:8)u(cid:8)2 =0, which implies u=0. Hence ψ(1) =ψ(2), i.e.,
2
the solution of (2.4) is unique.
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OPTIMAL ERROR ESTIMATES OF FINITE DIFFERENCE METHODS 107
Next, we prove the existence. For (j,k)∈T , rewrite equation (2.4) as
(cid:2) M(cid:3)
1
(3.12) iψn+1+τ − δ2 +V −ΩLh ψn+1+P =0,
jk 2 ∇ jk z jk jk
where P ∈X is defined as
M (cid:2) (cid:3)
1
(3.13) P =−iψn−1+2τβ|ψn |2ψn +τ − δ2 +V −ΩLh ψn−1.
jk jk jk jk 2 ∇ jk z jk
Consider the map G:ψ∗ ∈X →G(ψ∗)∈X defined as
M(cid:2) M (cid:3)
1
(3.14) G(ψ∗) =iψ∗ +τ − δ2 +V −ΩLh ψ∗ +P , (j,k)∈T .
jk jk 2 ∇ jk z jk jk M
We know that G is continuous from X to X . Noticing (3.4) in Lemma 3.1, we
M M
have
(3.15) Im(G(ψ∗),ψ∗)=(cid:8)ψ∗(cid:8)2+Im(P,ψ∗)≥(cid:8)ψ∗(cid:8)2−(cid:8)P(cid:8) (cid:8)ψ∗(cid:8) ,
2 2 2 2
which immediately implies
|(G(ψ∗),ψ∗)|
(3.16) lim =∞.
(cid:7)ψ∗(cid:7)2→∞ (cid:8)ψ∗(cid:8)2
Hence G : X → X is surjective [27] and there exists a solution ψn+1 ∈ X
M M M
satisfying G(ψn+1) = 0. Then ψn+1 satisfies the equation (2.4). The proof is
complete. (cid:3)
Define the local truncation error ηn ∈X of the SIFD method (2.4) with (2.2),
M
(2.3) and (2.5) for n≥1 as
(3.17)
(cid:2) (cid:3)
ηn :=iδ ψ(x ,y ,t )− −1δ2 −ΩLh+V ψ(xj,yk,tn−1)+ψ(xj,yk,tn+1)
jk t j k n 2 ∇ z jk 2
−β|ψ(x ,y ,t )|2ψ(x ,y ,t ), (j,k)∈T ,
j k n j k n M
and by noticing (2.3) for n=0 as
(cid:7) (cid:8)
1
(3.18) η0 :=iδ+ψ(x ,y ,0)− − δ2 +V −ΩLh ψ(1)
jk t j k 2 ∇ jk z jk
−β|ψ(1)|2ψ(1), (j,k)∈T ,
jk jk (cid:2)(cid:7) M (cid:8)
τ 1
ψ(1) =ψ (x ,y )−i − δ2 +V −ΩLh ψ (x ,y )
jk 0 j k 2 2 ∇ jk z 0 j k
(cid:3)
+β|ψ (x ,y )|2ψ (x ,y ) .
0 j k 0 j k
Then we have
Lemma 3.3 (Local truncation error). Assuming V(x)∈C(U), under assumption
(B), we have
T
(3.19) (cid:8)ηn(cid:8)∞ (cid:2)τ2+h2, 0≤n≤ τ −1, and (cid:8)δ∇+η0(cid:8)∞ (cid:2)τ +h.
In addition, assuming V(x)∈C1(U) and τ (cid:2)h, we have for 1≤n≤ T −1,
(cid:24) τ
τ2+h2, 1≤j ≤M −2, 1≤k ≤K−2,
(3.20) |δ+ηn |(cid:2)
∇ jk τ +h, j =0,M −1, or k =0,K−1.
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108 WEIZHUBAOANDYONGYONGCAI
Furthermore, assuming either Ω = 0 and V(x) = 0 or u ∈ C([0,T];H2(U)), we
0
have
T
(3.21) (cid:8)δ∇+ηn(cid:8)∞ (cid:2)τ2+h2, 1≤n≤ τ −1.
Proof. First,weprove(3.19)and(3.21)whenn=0. Rewritingψ(1) andthenusing
jk
Taylor’s expansion at (x ,y ,0), noticing (1.1) and (1.3), we get
j k
(3.22) (cid:5) (cid:6) (cid:2)(cid:7) (cid:8)
τ τ 1
ψ(1) =ψ x ,y , +i δ2 −V +ΩLh ψ (x ,y )
jk j k 2 2 2 ∇ jk z 0 j k
(cid:17) (cid:18) (cid:3)
ψ x ,y ,τ −ψ (x ,y )
−β|ψ (x ,y )|2ψ (x ,y )+i j k 2 0 j k
0 j k 0 j k τ/2
(cid:5) (cid:6) (cid:2) (cid:19) (cid:5) (cid:6) (cid:5) (cid:6)
τ τ h
=ψ x ,y , +i ∂ ψ x +hθ(2),y +∂ ψ x ,y +hθ(3)
j k 2 2 6 xxx 0 j jk k yyy 0 j k jk
(cid:5) (cid:5) (cid:6) (cid:5) (cid:6)(cid:6)(cid:20)
−3iΩ x ∂ ψ x ,y +hθ(4) −y ∂ ψ x +hθ(5),y
j yy 0 j k jk k xx 0 j jk k
(cid:5) (cid:6)(cid:3) (cid:5) (cid:6) (cid:17) (cid:18)
τ τ
+i ∂ ψ x ,y ,τθ(1) =ψ x ,y , +O τ2+τh , (j,k)∈T ,
4 tt j k jk j k 2 M
where θ(1) ∈ [0,1/2] and θ(2), θ(3), θ(4), θ(5) ∈ [−1,1] are constants. Similarly,
jk jk jk jk jk
using Taylor’s expansion at (x ,y ,τ/2) in (3.18), noticing (1.1) and (3.22), using
j k
triangle inequality and assumption (B), we get
|ηj0k|(cid:2)τ(cid:13)2(cid:8)∂tttψ(cid:8)L∞ +h2[(cid:8)∂xxxxψ(cid:8)L∞ +(cid:8)∂yyyyψ(cid:8)L∞ +(cid:8)∂xxxψ(cid:8)L∞ +(cid:8)∂yyyψ(cid:8)L∞(cid:14)]
+τ2 (cid:13)(cid:8)∂ttxxψ(cid:8)L∞ +(cid:8)∂ttyyψ(cid:8)L∞ +(cid:8)∂ttxψ(cid:8)(cid:14)L∞ +(cid:17)(cid:8)∂ttyψ(cid:8)L(cid:18)∞ +(cid:8)∂ttψ(cid:8)L∞ (cid:8)ψ(cid:8)2L∞
+τh (cid:8)ψ0(cid:8)W5,∞(U)+(cid:8)ψ(cid:8)2L∞ (cid:8)ψ0(cid:8)W3,∞(U) +O h4+τ4
(cid:2)τ2+h2, (j,k)∈T ,
M
where the L∞-norm means (cid:8)f(cid:8)L∞ :=sup0≤t≤T supx∈U|f(x,t)|. This immediately
implies (3.19) when n=0 as
(cid:8)η0(cid:8)∞ = max |ηj0k|(cid:2)τ2+h2.
(j,k)∈T0
M
Similarly, noticing τ (cid:2)h,
1
|δ+η0 | (cid:2) |η0 |(cid:2)τ +h, (j,k)∈T ,
∇ jk h jk M
which immediately implies (3.21) when n = 0. Now we prove (3.19), (3.20) and
(3.21)whenn≥1. UsingTaylor’sexpansionat(x ,y ,t )in(3.17),noticing(1.1),
j k n
using triangle inequality and assumption (B), we have
|ηjnk|(cid:2)h2[(cid:8)∂xxxxψ(cid:8)L∞ +(cid:8)∂yyyyψ(cid:8)L∞ +(cid:8)∂yyyψ(cid:8)L∞ +(cid:8)∂xxxψ(cid:8)L∞]
+τ2[(cid:8)∂tttψ(cid:8)L∞ +(cid:8)∂ttxxψ(cid:8)L∞ +(cid:8)∂ttyyψ(cid:8)L∞ +(cid:8)∂yttψ(cid:8)L∞ +(cid:8)∂xttψ(cid:8)L∞]
T
(cid:2)τ2+h2, (j,k)∈T , 1≤n≤ −1,
M τ
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