2016
DOI: 10.1002/num.22125
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A linearized, decoupled, and energy‐preserving compact finite difference scheme for the coupled nonlinear Schrödinger equations

Abstract: In this article, a decoupled and linearized compact finite difference scheme is proposed for solving the coupled nonlinear Schrödinger equations. The new scheme is proved to preserve the total mass and energy which are defined by using a recursion relationship. Besides the standard energy method, an induction argument together with an H1 technique are introduced to establish the optimal point‐wise error estimate of the proposed scheme. Without imposing any constraints on the grid ratios, the convergence order … Show more

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Cited by 19 publications
(10 citation statements)
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“…Suppose that the discrete mesh function { w n | n = 1, 2 , …, K; Kτ = T } satisfies recurrence formula wn+1wnC1τwn+1+C2τwn+Clτ, where C 1 , C 2 and C l ( l = 1, 2 , …, K ) are nonnegative constants. Then max1lKwnw0+τfalse∑l=1KCle2C1+C2T, where τ is small, such that ( C 1 + C 2 ) τ ⩽ ( K − 1)/2 K ( K > 1) . Lemma (). For any grid functions u ∈ X h , the following relations hold δxu2hu,δxuRδxu62δxu. …”
Section: Numerical Analysismentioning
confidence: 99%
“…Suppose that the discrete mesh function { w n | n = 1, 2 , …, K; Kτ = T } satisfies recurrence formula wn+1wnC1τwn+1+C2τwn+Clτ, where C 1 , C 2 and C l ( l = 1, 2 , …, K ) are nonnegative constants. Then max1lKwnw0+τfalse∑l=1KCle2C1+C2T, where τ is small, such that ( C 1 + C 2 ) τ ⩽ ( K − 1)/2 K ( K > 1) . Lemma (). For any grid functions u ∈ X h , the following relations hold δxu2hu,δxuRδxu62δxu. …”
Section: Numerical Analysismentioning
confidence: 99%
“…Due to the importance of two-component BECs in applications, a lot of attention have been paid to investigate the dynamic properties of CGP equations or coupled nonlinear Schrödinger (CNS) equations (when the external potential function V equals to a constant) numerically. A large amount of literatures provide various reliable and efficient strategies for solving these equations, including the finite difference method, finite element method, and spectral/pseudo-spectral method [7][8][9][10][11][12][13][14][15][16][17][18][19][20]. For a one-dimensional case, the existence and uniqueness for the ground states of two-component BECs with an internal atomic Josephson junction are studied in [7].…”
Section: Introductionmentioning
confidence: 99%
“…In [10], Sun and Zhao obtain the optimal convergence rate of the numerical solutions with the order of O(τ 2 + h 2 ) for the CNS equations by using the nonlinear finite difference scheme proposed in [9]. Consequently, the authors of [11,12] construct several conservative compact finite difference schemes for the CGP and CNS equations. Meanwhile, they establish the optimal point-wise error estimates and improve the convergence order to O(τ 2 + h 4 ).…”
Section: Introductionmentioning
confidence: 99%
“…There have been some numerical methods developed to compute their numerical solutions, for example, finite difference method, [4][5][6] compact finite difference scheme, 1,7-9 linearized energy-preserving compact finite difference scheme, 2 symplectic and multi-symplectic scheme, 10,11 splitting multi-symplectic method, 12,13 local energy-preserving method, 14 fourth order exponential time differencing method with local discontinuous Galerkin approximation, 15 energy-preserving Galerkin method. 3 We consider the following coupled nonlinear Schr€ odinger equations on the whole line juðx; tÞj 2 þ jvðx; tÞj 2 dx ¼ Z 1…”
Section: Introductionmentioning
confidence: 99%