In this section, we prove optimal error estimate for the CNFD method (6.7) with (6.8) and (6.9) in l2-norm, discrete H1-norm and l∞-norm. Let ψn ∈ XM K be the numerical solution of the CNFD method and en∈XM K be the error function.
Lemma 6.6 (Conservation of mass and energy) For the CNFD scheme (6.7) with (6.8) and (6.9), for any mesh size h >0, time stepτ >0 and initial data ψ0, it conserves the mass and energy in the discretized level, i.e.
kψnk22≡ kψ0k22, Eh(ψn)≡Eh(ψ0), n= 0,1,2, . . . . (6.63) Proof: Follow the analogous arguments of the CNFD method for the NLSE [46, 67] and we omit the details here for brevity.
Lemma 6.7 (Solvability of the difference equations) For any given ψn, there exists a solution ψn+1 of the CNFD discretization (6.7) with (6.8) and (6.9). In addition, assume
τ .h and either β ≥0 or β <0 with kψ0k22 < 1
|β|
1− Ωγ22
, under the Assumption (A),
there exists h0 >0 sufficiently small, when 0< h≤h0, the solution is unique.
Proof: First, we prove the existence of a solution of the CNFD discretization (6.7). In order to do so, for any given ψn∈XM K, we rewrite the equation (6.7) as
ψn+1/2 =ψn+iτ
2Fn(ψn+1/2), n= 0,1, . . . , (6.64) where Fn: XM K →XM K defined as
(Fn(u))jk=
−1
2δ∇2 +Vjk−ΩLhz
ujk+β
2(|2ujk−ψj,kn |2+|ψj,kn |2)ujk, (j, k)∈ TM K. Define the mapGn: XM K →XM K as
Gn(u) =u−ψn−iτ
2Fn(u), u∈XM K,
and it is easy to see that Gn is continuous fromXM K to XM K. Moreover, Re (Gn(u), u) =kuk22−Re(ψn, u)≥ kuk2(kuk2− kψnk2), u∈XM K, which immediately implies
kuklim2→∞
|(Gn(u), u)| kuk2
=∞.
Thus Gn is surjective. By using the Brouwer fixed point theorem (cf. [94]), it is easy to show that there exists a solution u∗ with Gn(u∗) = 0, which implies that there exists a solution ψn+1/2 to the problem (6.64) and thus the CNFD discretization (6.7) is solvable for any given ψn. In addition, for the solutionψn+1 to (6.7), using (6.63), we have
kδ∇+ψn+1k22≤C Eh(ψn+1) =C Eh(ψ0), n= 0,1, . . . ; (6.65) where when β ≥0, we have C = 2; and when β <0 with kψ0k22 < |β1|(1−Ωγ22), it comes from
Eh(ψ0) = Eh(ψn+1)≥ 1 2
1− Ω2
γ2
kδ∇+ψn+1k22−|β|
2 kδ+∇ψn+1k22ã kψn+1k22
= 1
2
1−Ω2 γ2
kδ+∇ψn+1k22−|β|
2 kδ∇+ψn+1k22ã kψ0k22
= |β| 2
1
|β|
1−Ω2 γ2
− kψ0k22
kδ∇+ψn+1k22.
Thus assumeh <1, whenβ ≥0 orβ <0 withkψ0k22 < 1
|β|
1−Ωγ22
, using (6.65) and the inverse inequality [145], we obtain
kψn+1k∞≤C|lnh| kδ∇+ψn+1k2≤C|lnh|Eh(ψ0), n= 0,1, . . . . (6.66) Next, we show the uniqueness of the solution of the CNFD scheme (6.7). For given ψn ∈XM K, suppose that there are two solutionsun+1∈XM K andvn+1∈XM K to (6.7).
From (6.66), we get
kun+1k∞≤C Eh(ψ0)|lnh|, kvn+1k∞≤C Eh(ψ0)|lnh|. (6.67) Denoting w:=un+1−vn+1∈XM K, from (6.7), we have
iwjk
τ =
−1
2δ∇2 +Vjk−ΩLhz
wjk+ ˆRjk, (j, k)∈ TM K, (6.68) where
Rˆjk= β
2(|un+1ij |2+|ψnjk|2)wjk+β
2(vn+1jk +ψnjk)(|un+1jk |2− |vjkn+1|2), (j, k)∈ TM K. Multiplying both sides of (6.68) with ¯wjk, summing for (j, k) ∈ TM K, and then taking imaginary parts, using (6.66) and (6.67), we have
kwk22 ≤τ C
kun+1k2∞+kvn+1k2∞+kψnk2∞
kwk22 ≤Cτ
Eh(ψ0) lnh2
kwk22. Thus under the assumption τ . h, there exists h0 > 0, when 0 < h ≤ h0, we have Cτ(lnh Eh(ψ0))2<1 which immediately implies
kwk2 =kun+1−vn+1k2= 0 =⇒ un+1=vn+1, i.e. the solution of CNFD (6.7) is unique.
Denote the local truncation error eηn∈XM K (n≥0) of the CNFD scheme (6.7) with (6.8) and (6.9) as
e
ηjkn : = iδt+ψ(xj, yk, tn)−
−1
2δ2∇−ΩLhz +Vjk+β
2 |ψ(xj, yk, tn+1)|2 +|ψ(xj, yk, tn)|2
×ψ(xj, yk, tn) +ψ(xj, yk, tn+1)
2 , (j, k)∈ TM K.(6.69) Then we have
Lemma 6.8 (Local truncation error) Assume V(x)∈L∞(U) and under the Assumption (B), we have
keηnk∞.τ2+h2, 0≤n≤ T
τ −1. (6.70)
In addition, assuming V(x)∈C1(U) andτ .h, we have for1≤n≤ Tτ −1
|δ∇+eηnjk|.
τ2+h2, 1≤j≤M −2,1≤k≤K−2, τ +h, j= 0, M−1, or k= 0, K−1.
(6.71)
In addition, if either Ω = 0 and ∂nV(x) = 0 or ψ∈C0([0, T];H02(U)), we have kδ∇+eηnk∞.τ2+h2, 1≤n≤ T
τ −1. (6.72)
Proof: Follow the analogous line for Lemma 6.3 and we omit it here for brevity.
Theorem 6.4 (l2-norm estimate) Assumeτ .h and eitherβ≥0orβ <0withkψ0k22<
1
|β|
1−Ωγ22
, under the Assumption (A) and (B), there existh0>0andτ0>0sufficiently small, when 0< h≤h0 and 0< τ ≤τ0, we have
kenk2.τ2+h2, kψnk∞≤√
2(1 +M1), 0≤n≤ T
τ. (6.73) Proof: Choose a smooth function α(ρ) (ρ≥0)∈C∞([0,∞)) defined as
α(ρ) =
1, 0≤ρ≤1,
∈[0,1], 1≤ρ≤2,
0, ρ≥2.
(6.74)
Denote M0 = 2(1 +M1)2 >0 and define FM0(ρ) =α
ρ M0
ρ, 0≤ρ <∞, thenFM0(ρ)∈C∞([0,∞)) and it is global Lipschitz, i.e.
|FM0(ρ1)−FM0(ρ2)| ≤CM0|√ρ1−√ρ2|, 0≤ρ1, ρ2 <∞. (6.75) Chooseφ0 =ψ0 ∈XM K and defineφn∈XM K (n= 0,1, . . .) as for (j, k)∈ TM K
iδt+φnjk=
−1
2δ2∇+Vjk−ΩLhz +β 2
FM0(|φn+1jk |2) +FM0 |φnjk|2
φn+1/2jk , (6.76)
where
φn+1/2jk = 1
2(φn+1jk +φnjk), (j, k)∈ TM K0 , n≥0.
In fact,φncan be viewed as another approximation ofψ(x, tn). Define the ‘error’ function ˆ
en∈XM K
ˆ
enjk:=ψ(xj, yk, tn)−φnjk, (j, k)∈ TM K0 , n≥0, and the local truncation error ˆηn∈XM K of the scheme (6.76) as ˆ
ηjkn := iδ+t ψ(xj, yk, tn)−
−1
2δ∇2 −ΩLhz+Vjk+β 2
FM0(|ψ(xj, yk, tn+1)|2) (6.77) +FM0(|ψ(xj, yk, tn)|2)
×ψ(xj, yk, tn) +ψ(xj, yk, tn+1)
2 , (j, k)∈ TM K, n≥0.
Similar as Lemma 6.8, we can prove
kηˆnk∞.τ2+h2, 0≤n≤ T τ. Subtracting (6.77) from (6.76), we obtain
iδt+ˆenj,k =
−1
2δ2∇+Vjk−ΩLhz
ˆ
en+1/2jk +β 2
FM0(|φn+1jk |2) +FM0(|φnjk|2) ˆ en+1/2jk +β
4(ψ(xj, yk, tn+1) +ψ(xj, yk, tn)) ˆξnjk+ ˆηjkn, (j, k)∈ TM K, n≥0,(6.78) where ˆξn∈XM K defined as
ξˆjkn =FM0(|φn+1jk |2)+FM0(|φnjk|2)−FM0(|ψ(xj, yk, tn+1)|2)−FM0(|ψ(xj, yk, tn)|2), (j, k) ∈ TM K0 . This together with (6.75) implies
β
4 (ψ(xj, yk, tn+1) +ψ(xj, yk, tn)) ˆξjkn .C
|eˆn+1jk |+|eˆnjk|
, (j, k)∈ TM K0 . Multiplying both sides of (6.78) with ˆen+1jk + ˆenjk, summing for (j, k)∈ TM K, taking imag- inary part and applying the Cauchy inequality, we obtain
keˆn+1k22− kˆenk22 . τ |ηˆn|2∞+C(keˆn+1k22+kˆenk22) . τ
(h2+τ2)2+ (keˆn+1k22+keˆnk22)
, 0≤n≤ T τ −1.
Then there exists τ0 > 0 sufficiently small, when 0 < τ ≤ τ0, applying the discrete Gronwall inequality [46, 67, 95], we get
keˆnk2.τ2+h2, 0≤n≤ T τ.
Applying the inverse inequality in 2D, we have keˆnk∞. 1
hkeˆnk2 .h+ τ2
h .h, 0≤n≤ T
τ, (6.79)
which implies
kφnk∞≤ kΠhψ(tn)k∞+kˆenk∞≤
√M0
2 +Ch, 0≤n≤ T τ. Thus under the assumption τ .h, there exists h0 >0, when 0< h≤h0, we have
kφnk∞≤
√M0
2 +
√M0
2 =p
M0 =⇒ kφnk2∞≤M0, 0≤n≤ T
τ. (6.80) Therefore, the discretization (6.76) collapses exactly to the CNFD discretization (6.7) with (6.8) and (6.9), i.e.
ψn=φn, en= ˆen, 0≤n≤ T τ. This together with (6.79) and (6.80) complete the proof.
Again, combining Theorem 6.4 and Lemmas 6.7 and 6.8, we are now ready to prove the main Theorem 6.2.
Proof of Theorem 6.2: As in the proof of Theorem 6.1, we only prove the optimal convergence under the Assumption (A) and (B) with either Ω = 0 and ∂nV(x) = 0 or ψ∈C0([0, T];H02(U)). Subtracting (6.69) from (6.7), we get
iδt+enjk =
−1
2δ2∇+Vjk−ΩLhz
en+1/2jk +ξenjk+ηejkn, (j, k)∈ TM K, n≥0,(6.81) where ˜ξn∈XM K defined as
ξejkn = β 2
h
enjkψ(xj, yk, tn) +ψjknenjk+en+1jk ψ(xj, yk, tn+1) +ψn+1jk en+1jk i ψn+1/2jk +β
2(|ψ(xj, yk, tn)|2+|ψ(xj, yk, tn+1)|2)en+1/2jk , (j, k)∈ TM K. Again, rewrite (6.81) as
en+1−en=−iτ e
χn+ξen+eηn
, n≥0, (6.82)
where χen∈XM K defined as e
χnjk=
−1
2δ2∇+Vjk−ΩLhz
en+1/2jk , (j, k)∈ TM K, n≥0.
Multiplying both sides of (6.81) with en+1jk −enjk, summing for (j, k) ∈ TM K, noticing (6.22), (6.23) and (6.82), taking real parts, we obtain
E(en+1)− E(en) = −2 ReD
ξen+eηn, en+1−enE
= −2 ReD
ξen+eηn,−iτ(χen+ξen+eηn)E
= 2τ ImD
ξen+ηen,χenE
, 0≤n≤ T τ −1.
Similar as those in the proof of Theorem 6.1, we can prove
ImD
ξen+ηen,χenE .(h2+τ2)2+E(en+1) +E(en), 0≤n≤ T τ −1.
Combining the above two inequalities, we get E(en+1)− E(en).τ
(τ2+h2)2+E(en+1) +E(en)
, 0≤n≤ T
τ −1. (6.83) Then there exists τ0 >0 sufficiently small, when 0< τ ≤τ0, using the discrete Gronwall inequality [46, 67, 95] and noticing e0 = 0 and E(e0) = 0, we get
E(en).(τ2+h2)2, 0≤n≤ T τ,
which immediately implies (6.20). If we only have Assumption (A) and (B) without further assumption, the convergence rate will beO(h3/2+τ3/2). The proof is the same as
in Theorem 6.1, and we omit it here.
Similarly, from Theorem 6.2 and using the inverse inequality [145], we get immediately the error estimate inl∞-norm for the CNFD method as
Lemma 6.9 (l∞-norm estimate) Under the same conditions of Theorem 6.2 and assume h <1, with Assumption (A) and (B), we have the following error estimate for the CNFD
kenk∞.
(h3/2+τ3/2)|ln(h)|, d= 2,
h+τ, d= 3.
In addition, if either Ω = 0 and ∂nV(x) = 0 or ψ∈C0([0, T];H02(U)), we have kenk∞ .
(h2+τ2)|ln(h)|, d= 2, h3/2+τ3/2, d= 3.
Remark 6.2 If the cubic nonlinear term β|ψ|2ψ in (6.1) is replaced by a general nonlin- earity f(|ψ|2)ψ, the numerical discretization CNFD and its error estimates in l2-norm, l∞-norm and discreteH1-norm are still valid provided that the nonlinear real-valued func- tion f(ρ)∈C3([0,∞)).