Lyapunov-Sylvester Operators for Numerical Solutions of NLS Equation
aa r X i v : . [ m a t h . NA ] N ov Lyapunov-Sylvester Operators for NumericalSolutions of NLS Equation
Riadh CHTEOUI
D´epartement de Math´ematiques, Facult´e des Sciences de Monastir, 5019 Monastir,Tunisia.
Anouar BEN MABROUK ∗ , Computational Mathematics Lab, D´epartement de Math´ematiques, Facult´e desSciences, 5019 Monastir, Tunisia.
Hichem OUNAIES
D´epartement de Math´ematiques, Facult´e des Sciences de Monastir, 5019 Monastir,Tunisia.
Abstract
In the present paper a numerical method is developed to approximate the solution oftwo-dimensional NLS equation in the presence of a singular potential. The methodleads to Lyapunov-Syslvester algebraic operators that are shown to be invertibleusing original topological and differential calculus issued methods. The numeri-cal scheme is proved to be consistent, convergent and stable using the based onLyapunov criterion, lax equivalence theorem and the properties of the Lyapunov-Syslvester operators.
Key words:
NLS equation, Finite-difference scheme, Stability analysis, Lyapunovcriterion, Consistency, Convergence, Error estimates, Lyapunov-Sylvester operator.
PACS: ∗ Corresponding Author
Email addresses: [email protected] (Riadh CHTEOUI), [email protected] (Anouar BEN MABROUK), [email protected] (Hichem OUNAIES). D´epartement de Math´ematiques, Institut Sup´erieur de Math´amatiques Ap-pliqu´ees et Informatique de Kairouan, Avenue Assad Ibn Al-Fourat, Kairouan 3100,Tunisia.
Preprint submitted to Elsevier 26 May 2018
Introduction
The Schr¨odinger equation is widely studied from both numerical and theoret-ical points of view. This is due to its relation to the modeling of real physicalphenomena such as Newton’s laws and conservation of energy in classical me-chanics, behaviour of dynamical systems, the description of a particle in anon-relativistic setting in quantum mechanics, etc. The Schr¨odinger’s linearequation states that ∆ ψ + 8 π m ~ ( E − V ( x )) ψ = 0 , where ψ is the Schr¨odinger wave function, m is the mass, ~ denotes Planck’sconstant, E is the energy, and V is the potential energy. However, in thenonlinear case, the structure of the nonlinear Schr¨odinger equation is morecomplicated. This equation is a prototypical dispersive nonlinear partial dif-ferential equation related Bose-Einstein condensates and nonlinear optics ([9]),propagation of electric fields in optical fibers ([21], [27]), self-focusing and col-lapse of Langmuir waves in plasma physics ([33]), behaviour of rogue waves inoceans ([29]).The nonlinear Schr¨odinger equation is also related to electromagnetic, ferro-magnetic fields as well as magnums, high-power ultra-short laser self-channellingin matter, condensed matter theory, dissipative quantum mechanics, ([2]), filmequations, etc (See [1], [31]).Based upon the analogy between mechanics and optics, Schr¨odinger estab-lished the classical derivation of his equation. By developing a perturbationmethod, he proved the equivalence between his wave mechanics equation andand Heisenberg’s matrix one, and thus introduced the time dependent versionstated hereafter with a cubic nonlinearity i ~ ψ t = − ~ m ∆ ψ + V ( x ) ψ − γ | ψ | ψ in R N ( N ≥ . (1)In [18] and [28] the potential V is assumed to be bounded with a non-degenerate critical point at x = 0. More precisely, V belongs to the class V a ,for some real parameter a (See [25]). With suitable assumptions it is provedin [28] a Lyapunov-Schmidt type reduction the existence of standing wavesolutions of problem (1), of the form ψ ( x, t ) = e − iEt/ ~ u ( x ) . (2)Hence, the nonlinear Schr¨odinger equation (1) is reduced to the semilinearelliptic equation − ~ m ∆ u + ( V ( x ) − E ) u = | u | u . y = ~ − x and replacing y by x we get − ∆ u + 2 m ( V ~ ( x ) − E ) u = | u | u in R N , (3)where V ~ ( x ) = V ( ~ x ).If for some ξ ∈ R N \ { } , V ( x + sξ ) = V ( x ) for all s ∈ R , equation (1) isinvariant under the Galilean transformation ψ ( x, t ) exp (cid:18) iξ · x/ ~ − i | ξ | t/ ~ (cid:19) ψ ( x − ξt, t ) . Thus, in this case, standing waves reproduce solitary waves traveling in thedirection of ξ .The present paper is devoted to the development of a numerical method basedon two-dimensional finite difference scheme to approximate the solution of thenonlinear Schr¨odinger (NLS) equation in R written on the form iu t + ∆ u + u − | u | − θ u = 0 , ∈ Ω × ( t , + ∞ ) (4)with the initial and boundary conditions u ( x, y, t ) = u ( x, y ) and ∂ u∂ n ( x, y, t ) = 0 , ( x, y, t ) ∈ Ω × ( t , + ∞ ) . (5)We consider a rectangular domain Ω =] L , L [ × ] L , L [ in R and t a realparameter fixed as the initial time, u t is the first order partial derivative intime, ∆ = ∂ ∂ x + ∂ ∂ y is the Laplace operator in R . ∂∂n is the outward normalderivative operator along the boundary ∂ Ω. Finally, u and u are complexevalued functions.In [10], the stationary solutions of problem (4) has been studied using directmethods issued from the equation on the whole space. Existence, unique-ness, classification and properties of the solutions have been investigated. Itis proved that three attractive zones or three classes of stationary solutionsexists; there are solutions oscillating around 0 with supports being compact,there are solutions oscillating around ± C . Let Ω =] L , L [ × ] L , L [ ⊂ R and for J ∈ N ∗ , denote h = L − L J for thespace step, x j = L + jh and y m = L + mh for all ( j, m ) ∈ I = { , , . . . , J } .Let l = ∆ t be the time step and t n = t + nl , n ∈ N be the discrete timegrid. For ( j, m ) ∈ I and n ≥ u nj,m will be the net function u ( x j , y m , t n )and U nj,m the numerical solution. The following discrete approximations willbe applied for the different differential operators involved in the problem. Fortime derivatives, we set u t U n +1 j,m − U n − j,m l and for space derivatives, we shall use u x U nj +1 ,m − U nj − ,m h and u y U nj,m +1 − U nj,m − h for first order derivatives and for second order ones we apply the estimations∆ u ∆ U nj,m ( α n , β n , γ n ) = ∆ " α n U n +1 j,m + β n U nj,m + γ n U n − j,m where ∆ U nj,m = U nj +1 ,m − U nj,m + U nj − ,m h + U nj,m +1 − U nj,m + U nj,m − h . where α n , β n et γ n are sequences in [0,1] such that α n + β n + γ n = 1.By replacing the derivatives of u with their approximations, the equation (4)4ields i U n +1 j,m − U n − j,m l + ∆ " α n U n +1 j,m + β n U nj,m + γ n U n − j,m + d f nj,m = 0where d f nj,m = f ( U nj,m ) + f ( U n − j,m )2 where we design by f ( u ) = u − | u | − θ u . Wethen obtain i U n +1 j,m − U n − j,m l + α n U n +1 j − ,m − U n +1 j,m + U n +1 j +1 ,m + U n +1 j,m − − U n +1 j,m + U n +1 j,m +1 h + β n U nj − ,m − U nj,m + U nj +1 ,m + U nj,m − − U nj,m + U nj,m +1 h + γ n U n − j − ,m − U n − j,m + U n − j +1 ,m + U n − j,m − − U n − j,m + U n − j,m +1 h + d f nj,m = 0 . Denote next σ = 2 lh . We obtain iU n +1 j,m − iU n − j,m + σα n U n +1 j − ,m − U n +1 j,m + U n +1 j +1 ,m + U n +1 j,m − − U n +1 j,m + U n +1 j,m +1 + σβ n U nj − ,m − U nj,m + U nj +1 ,m + U nj,m − − U nj,m + U nj,m +1 + σγ n U n − j − ,m − U n − j,m + U n − j +1 ,m + U n − j,m − − U n − j,m + U n − j,m +1 + 2 l d f nj,m = 0 . By setting ϕ n = i − σα n ψ n = i + 4 σγ n σα n U n +1 j − ,m + ϕ n U n +1 j,m + σα n U n +1 j +1 ,m + σα n U n +1 j,m − + ϕ n U n +1 j,m + σα n U n +1 j,m +1 + σβ n U nj − ,m − σβ n U nj,m + σβ n U nj +1 ,m + σβ n U nj,m − − σβ n U nj,m + σβ n U nj,m +1 + σα n U n − j − ,m − ψ n U n − j,m + σα n U n − j +1 ,m + σα n U n − j,m − − ψ n U n − j,m + σα n U n − j,m +1 + 2 l d f nj,m = 0 .
5r in vector form (cid:18) σα n ϕ n σα n (cid:19) U n +1 j − ,m U n +1 j,m U n +1 j +1 ,m + (cid:18) U n +1 j,m − U n +1 j,m U n +1 j,m +1 (cid:19) σα n ϕ n σα n + σβ n (cid:18) − (cid:19) U nj − ,m U nj,m U nj +1 ,m + σβ n (cid:18) U nj,m − U nj,m U nj,m +1 (cid:19) − + (cid:18) σγ n ψ n σγ n (cid:19) U n − j − ,m U n − j,m U n − j +1 ,m + (cid:18) U n − j,m − U n − j,m U n − j,m +1 (cid:19) σγ n ψ n σγ n + 2 l d f nj,m = 0 . Now, we exploit the boundary conditions which can be resumed in the follow-ing cases of the parameters j, m . Indeed, by setting in the previous equation j = m = 0 and using the approximations of boundary conditions we obtain (cid:18) ϕ n σα n (cid:19) U n +10 , U n +11 , + (cid:18) U n +10 , U n +10 , (cid:19) ϕ n σα n + σβ n (cid:18) − (cid:19) U n , U n , + σβ n (cid:18) U n , U n , (cid:19) − + (cid:18) ψ n σγ n (cid:19) U n − , U n − , + (cid:18) U n − , U n − , (cid:19) ψ n σγ n + 2 l d f n , = 0 . For j = 0 and m = J , we obtain as previously, (cid:18) ϕ n σα n (cid:19) U n +10 ,J U n +11 ,J + (cid:18) U n +10 ,J − U n +10 ,J (cid:19) σα n ϕ n + σβ n (cid:18) − (cid:19) U n ,J U n ,J + σβ n (cid:18) U n ,J − U n ,J (cid:19) − + (cid:18) ψ n σγ n (cid:19) U n − ,J U n − ,J + (cid:18) U n − ,J − U n − ,J (cid:19) σγ n ψ n + 2 l d f n ,J = 0 . j = 0 and 1 ≤ m ≤ J −
1, we get (cid:18) ϕ n σα n (cid:19) U n +10 ,m U n +11 ,m + (cid:18) U n +10 ,m − U n +10 ,m U n +10 ,m +1 (cid:19) σα n ϕ n σα n + σβ n (cid:18) − (cid:19) U n ,m U n ,m + σβ n (cid:18) U n ,m − U n ,m U n ,m +1 (cid:19) − + (cid:18) ψ n σγ n (cid:19) U n − ,m U n − ,m + (cid:18) U n − ,m − U n − ,m U n − ,m +1 (cid:19) σγ n ψ n σγ n + 2 l d f n ,m = 0 . Now, by setting j = J and m = 0 and using the approximations of boundaryconditions we obtain (cid:18) σα n ϕ n (cid:19) U n +1 J − , U n +1 J, + (cid:18) U n +1 J, U n +1 J, (cid:19) ϕ n σα n + σβ n (cid:18) − (cid:19) U nJ − , U nJ, + σβ n (cid:18) U nJ, U nJ, (cid:19) − + (cid:18) σγ n ψ n (cid:19) U n − J − , U n − J, + (cid:18) U n − J, U n − J, (cid:19) ψ n σγ n + 2 l d f nJ, = 0 . For j = J and m = J , we obtain (cid:18) σα n ϕ n (cid:19) U n +1 J − ,J U n +1 J,J + (cid:18) U n +1 J,J − U n +1 J,J (cid:19) σα n ϕ n + σβ n (cid:18) − (cid:19) U nJ − ,J U nJ,J + σβ n (cid:18) U nJ,J − U nJ,J (cid:19) − + (cid:18) σγ n ψ n (cid:19) U n − J − ,J U n − J,J + (cid:18) U n − J,J − U n − J,J (cid:19) σγ n ψ n + 2 l d f nJ,J = 0 . j = J and 1 ≤ m ≤ J − (cid:18) σα n ϕ n (cid:19) U n +1 J − ,m U n +1 J,m + (cid:18) U n +1 J,m − U n +1 J,m U n +1 J,m +1 (cid:19) σα n ϕ n σα n + σβ n (cid:18) − (cid:19) U nJ − ,m U nJ,m + σβ n (cid:18) U nJ,m − U nJ,m U nJ,m +1 (cid:19) − + (cid:18) σγ n ψ n (cid:19) U n − J − ,m U n − J,m + (cid:18) U n − J,m − U n − J,m U n − J,m +1 (cid:19) σγ n ψ n σγ n + 2 l d f nJ,m = 0 . Next, for 1 ≤ j ≤ J − m = 0, we have (cid:18) σα n ϕ n σα n (cid:19) U n +1 j − , U n +1 j, U n +1 j +1 , + (cid:18) U n +1 j, U n +1 j, (cid:19) ϕ n σα n + σβ n (cid:18) − (cid:19) U nj − , U nj, U nj +1 , + σβ n (cid:18) U nj, U nj, (cid:19) − + (cid:18) σγ n ψ n σγ n (cid:19) U n − j − , U n − j, U n − j +1 , + (cid:18) U n − j, U n − j, (cid:19) ψ n σγ n + 2 l d f nj, = 0 . ≤ j ≤ J − m = J , we obtain (cid:18) σα n ϕ n σα n (cid:19) U n +1 j − ,J U n +1 j,J U n +1 j +1 ,J + (cid:18) U n +1 j,J − U n +1 j,J (cid:19) σα n ϕ n + σβ n (cid:18) − (cid:19) U nj − ,J U nj,J U nj + ,J + σβ n (cid:18) U nj,J − U nJ,J (cid:19) − + (cid:18) σγ n ψ n σγ n (cid:19) U n − j − ,J U n − j,J U n − j +1 ,J + (cid:18) U n − j,J − U n − j,J (cid:19) σγ n ψ n + 2 l d f nj,J = 0 . In the matrix form we obtain L A n ( U n +1 ) + L B n ( U n ) + L C n ( U n − ) + F n = 0 , (6)where U n = (cid:16) U nj,m (cid:17) is the matrix of the numerical solution. A n , B n and C n are ( J + 1) × ( J + 1) tri-diagonal matrices with respective coefficients A n ( j, j ) = ϕ n ,A n ( j, j −
1) = A n ( j, j + 1) = σα n ,A n (0 ,
1) = A n ( J, J −
1) = 2 σα n .B n ( j, j ) = − σβ n ,B n ( j, j −
1) = B n ( j, j + 1) = σβ n ,B n (0 ,
1) = B n ( J, J −
1) = 2 σβ n .C n ( j, j ) = ψ n ,C n ( j, j −
1) = C n ( j, j + 1) = σγ n ,C n (0 ,
1) = C n ( J, J −
1) = 2 σγ n . and F n = (2 l d f nj,m ). Finally, for a matrix W ∈ M J +1 ( C ), L W is the so-calledLyapunov operator defined on M J +1 ( C ) by L W ( X ) = W X + XW T , for all X . Thus, the discrete scheme leads to a Lyapunov algebraic recursive system.We now state the first result on the solvability of the numerical scheme. Theorem 2.1
The system (6) is uniquely solvable whenever U and U areknown.
9n [5], the authors have transformed the Lyapunov operator obtained from thediscretization method into a standard linear operator acting on one columnvector by juxtaposing the columns of the matrix X horizontally which leadsto an equivalent linear operator characterized by a fringe-tridiagonal matrix.We used standard computation to prove the invertibility of such an operator.Here. we do not apply the same computations as in [5], but we develop differentarguments. We will instead apply a differential calculus and topology technique(See [22] for example) to prove theorem 2.1. Lemma 2.1
Let E be a finite dimensional ( R or C ) vector space and (Φ n ) n bea sequence of endomorphisms converging uniformly to an invertible endomor-phism Φ . Then, there exists n such that, for any n ≥ n , the endomorphism Φ n is invertible. Indeed, consider the set
Isom ( E ) of isomorphisms on E . It is regarded as thereciprocal image det − ( C ∗ ) with the determinant function. As this function iscontinuous, thus it consists of an open set in the set L ( E ) of endomorphismsof E . Thus, as Φ ∈ Isom ( E ) there exists a ball B (Φ , r ) ⊂ Isom ( E ). Theelements Φ n are in this ball for large values of n . So these are invertible.Assume now that l = o ( h ε ), with ε > A and W will satisfy as h −→ A n j, j = i − α n lh −→ i ∀ ≤ j ≤ J. For 1 ≤ j ≤ J − A j,j − = A j,j +1 = A , A J,J − α n lh −→ . Next, observing that for all X in the space M ( J +1) ( C ), k ( L A n − iI )( X ) k = k ( A n − i I ) X + X ( A Tn − i I ) k≤ k A − I kk X k , it results that kL A n − iI k ≤ Ch ε . (7)Consequently, the Lyapunov endomorphism L A n converges uniformly to theisomorphism iI as h goes towards 0 and l = o ( h ε ) with ε >
0. Using Lemma2.1, the operator L A n is invertible for h small enough. Hence, Theorem 2.1 isproved. 10 Consistency, Stability and Convergence
The consistency of the proposed method is done by evaluating the local trun-cation error arising from the discretization of the system iu t + ∆ u + u − | u | − θ u = 0 , ∈ Ω × ( t , + ∞ )The principal part is L ( x, y, t ) = i l ∂ u∂t + ( α − γ ) l ∆ u t + α + γ l ∆ u tt + β h
12 ( ∂ u∂x + ∂ u∂y ) (8) Lemma 3.1 • α = γ and l = o ( h ) the scheme is consistent with order ( h + l ) . • α = γ the scheme is unconditionally consistent with order ( h + l ) . It is clear that the two operators L u tend toward 0 as l and h tend to 0,which ensures the consistency of the method. Furthermore, the method maybe always chosen to be consistent with an order 2 in time and space.We now prove the stability of the method by applying the Lyapunov criterionwhich states that a linear system L ( x n +1 , x n , x n − , . . . ) = 0 is stable in thesense of Lyapunov if for any bounded initial solution x the solution x n remainsbounded for all n ≥
0. Here, we will precisely prove the following result.
Lemma 3.2 P n : The solution U n is bounded independently of n whenever theinitial solution U is bounded. We will proceed by induction on n . Assume firstly that k U k ≤ η for some η positive. Using equation (6), we obtain kL A n ( U n +1 ) k ≤ kL B n k . k U n k + kL C n k . k U n − k + k F n − k + k F n k l = o ( h ε +2 ) small enough, ε >
0, we have for h −→ α n , β n and γ n are bounded). B n ( j, j ) = − σβ n ∼ Ch ε −→ ,B n ( j, j −
1) = B n ( j, j + 1) = σβ n ∼ Ch ε −→ ,B n (0 ,
1) = B n ( J, J −
1) = 2 σβ n ∼ Ch ε −→ . n ( j, j ) = ψ n ∼ i Ch ε −→ i ,C n ( j, j −
1) = C n ( j, j + 1) = σγ n ∼ Ch ε −→ ,C n (0 ,
1) = C n ( J, J −
1) = 2 σγ n ∼ Ch ε −→ . As a consequence, for h small enough, kL B n k ≤ k B n k ≤ Ch ε , (10)and kL C n k ≤ k C n k ≤
12 + Ch ε , (11)We shall next use the following lemma deduced from (7). Lemma 3.3
For h small enough, it holds for all X ∈ M ( J +1) ( R ) that k X k ≤ (1 − Ch ε ) k X k ≤ kL A n ( X ) k ≤ (1 + Ch ε ) k X k ≤ k X k . Indeed, recall that equation (7) affirms that kL A n − iI k ≤ Ch ε for someconstant C >
0. Consequently, for any X we get(1 − Ch ε ) k X k ≤ kL A n ( X ) k ≤ (1 + Ch ε ) k X k . For h ≤ C /ε , we obtain12 ≤ (1 − Ch ε ) < (1 + Ch ε ) ≤ k U n +1 k ≤ k (1+2 Ch ε ) k U n k +2(1+ Ch ε ) k U n − k + k U n − k − θ + k U n k − θ . (12)For n = 0, this implies that k U k ≤ k (1 + 2 Ch ε ) k U k + 2(1 + Ch ε ) k U − k + k U − k − θ + k U k − θ . (13)Using the discrete approximation U − = U − il (∆ u + f ( u ))and the fact that u is sufficiently regular and thus bounded on the domainΩ, we get k U − k ≤ k U k + Cl ≤ k U k + Ch ε . (14)Hence, equation (13) yields that k U k ≤ (3 + Ch ε ) k U k + Ch ε (1 + Ch ε ) + 2( k U k + Ch ε ) − θ . (15)12ow, the Lyapunov criterion for stability states exactly that ∀ ε > , ∃ η > s.t ; k U k ≤ η ⇒ k U n k ≤ ε, ∀ n ≥ . (16)For n = 1 and k U k ≤ ε , we seek an η > k U k ≤ η . Indeed, using(15), this means that, it suffices to find η such that(3 + Ch ε ) k U k + Ch ε (1 + Ch ε ) + 2( k U k + Ch ε ) − θ < ε. (17)Choosing h small enough ( h ≤ ηC ), we seek eta such that2 η ( η + 2) + 2 − θ η − θ < ε. (18)which is possible as the quantity at the left hand tends to 0 when η → U k is bounded for k = 1 , , . . . , n (by ε ) whenever U is bounded by η and let ε >
0. We shall prove that it is possible to choose η satisfying k U n +1 k ≤ ε . Indeed, from (12), we have k U n +1 k ≤ (3 + Ch ε ) ε + 2 ε − θ . (19)So, one seeks, ε for which (3 + Ch ε ) ε + 2 ε − θ < ε which is always possible.Next, the convergence is a consequence of the well known Lax-Richtmyerequivalence theorem, which states that for consistent numerical approxima-tions, stability and convergence are equivalent. Recall here that we have al-ready proved in (8) that the used scheme is consistent. Next, Lemma 3.2,Lemma 3.3 and equation (16) yields the stability of the scheme. Consequently,the Lax equivalence Theorem guarantees the convergence. So as the followingLemma. Lemma 3.4
As the numerical scheme is consistent and stable, it is then con-vergent.
In this section we propose to develop a numerical example to illustrate theefficiency of the numerical scheme proposed and studied previously. It con-sists of a model of interaction of two particles or two waves. We consider the13nhomogeneous problem i ∂u∂t + ∆ u + u − | u | − θ u = g ( x, y, t ) in Q ,u ( x, y,
0) = v ( x ) v ( y ) in Ω ,∂u∂n ( x, y, t ) = 0 on ∂ Ω × [0 , T ] (20)where Ω =] − L , L [ , Q = Ω × [0 , T ] , v ( x ) = cos (cid:18) π L x (cid:19) and g ( x, y, t ) = exp − iπ L t ! (cid:20) π L ( v ( x ) + v ( y )) + v ( x ) v ( y ) − v − θ ( x ) v − θ ( y ) (cid:21) . The exact solution is u ( x, y, t ) = exp − iπ L t ! cos (cid:18) π L x (cid:19) cos (cid:18) π L y (cid:19) . In the following tables, numerical results are provided. We computed for dif-ferent space and time steps the discrete L -error estimates defined as follows. k X k = N X i,j =1 | X ij | / for a matrix X = ( X ij ) ∈ M N +2 ( C ). Denote u n the net function u ( x, y, t n )and U n the numerical solution. The discrete L -error is Er = max n k U n − u n k (21)on the grid ( x i , y j ), 0 ≤ i, j ≤ J + 1. We compute also the relative errorbetween the exact solution and the numerical one as Relative Er = max n k U n − u n k k u n k (22)on the same grid.The domain Ω is chosen to be Ω =] − π, π [. The time interval is [0 ,
1] for achoice t = 0 and T = 1. The following results are obtained for different valuesof h , l and for θ = 14 . We choed finally the barycenter calibrating parameters α n , β n and γ n to be α n = 14 + 12 n +3 β n = 14 − n +3 and γ n = 12 . able 1.J l log( l ) / log( h ) Er Relative Er Er/ ( l + h )10 1/100 -20.15 7 , . − , . −
16 1/120 19.81 3 , . − , . −
20 1/200 11.40 1 , . − , . −
24 1/220 8.33 1 , . − , . −
30 1/280 6.47 8 , . − , . −
40 1/400 5.17 4 , . − , . −
50 1/500 4.50 3 , . − , . −
15s we see in the table, the numerical scheme converges with a convergencerate of ( l + h ). Notice from the last column that the quantity Er ( l + h ) isof the order of 10 − even in the case where the hypothesis l = o ( h ε ) is notsatisfied (line 1 of the table). In the 3rd column of the table, we notice thecontribution of approximate solution to the exact solution. A relative error of4% meaning that the ratio U numericalU exact is of the order of 1 ± . Remark 4.1
The barycenter parameters α n , β n and γ n are applied to cali-brates the position of the approximated solution relatively to the exact one.These parameters affect surely the numerical solution as well as the error esti-mates. In existing works such as [4], [5], [6], [7], [8] these are constants. Here.we adopted instead variable coefficients which may be generated by randomprocedures. These calibrations permits the use of implicit/explicit schemes byusing suitable values. For example for α n = γ n = and β n = 0 , the barycentreestimation becomes U n ( α n , β n , γ n ) = U n +1 + U n − which is an implicit estimation that guarantees an error of order in time. This paper investigated the solution of the well-known NLS equation in two-dimensional case by applying a two-dimensional finite difference discretization.The continuous problem is firstly recasted into an algebraic discrete systeminvolving Lyapunov-Syslvester matrix terms by using a full time-space dis-cretization. Solvability, consistency, stability and convergence are then estab-lished by applying Lax-Richtmyer equivalence theorem and Lyapunov stabilityand by examining the Lyapunov-Sylvester operators. The method was finallyimproved by developing a numerical example issued from 2-particles interac-tion. It was shown to be efficient by means of error estimates.
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