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

Bao, Weizhu; Cai, Yongyong

doi:10.1090/s0025-5718-2012-02617-2

Cited by 155 publications

(161 citation statements)

References 43 publications

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“…This confirms our error estimates (2.22) and (2.23) for SIFD. Table 5.3 presents the errors of SIFD at the degeneracy regime for α = 2 in the regime τ ∼ ε 2 , and resp., for α = 0 in the regime τ ∼ ε 3 , predicted by our error estimates. The results clearly demonstrate that SIFD converges at O(h 2 + τ ) and O(h 2 + τ 2/3 ) for α = 2 and α = 0, respectively.…”

Section: Lemma 34 Under the Assumptions In Theoremsupporting

confidence: 64%

“…However, for such a scheme, at each time step, a fully nonlinear system has to be solved very accurately, which may be very time-consuming, especially in 2D and 3D. In fact, if the nonlinear system is not solved very accurately, the numerical solution computed doesn't conserve the energy and/or mass exactly [3]. This motivates us also to consider the following SIFD discretization for NLSW in which at each time step only a linear system needs to be solved and it usually can be solved by fast direct Poisson solver.…”

Section: Finite Difference Schemes and Main Resultsmentioning

confidence: 99%

“…(See [3] for a discussion on the NLS case.) Thus their proof cannot be extended to either higher dimensions (2D or 3D) or nonconservative schemes.…”

Section: Introductionmentioning

confidence: 99%

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Uniform Error Estimates of Finite Difference Methods for the Nonlinear Schrödinger Equation with Wave Operator

Bao¹,

Cai²

2012

SIAM J. Numer. Anal.

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156

117

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Abstract. We establish uniform error estimates of finite difference methods for the nonlinear Schrödinger equation (NLS) perturbed by the wave operator (NLSW) with a perturbation strength described by a dimensionless parameter ε (ε ∈ (0, 1]). When ε → 0 + , NLSW collapses to the standard NLS. In the small perturbation parameter regime, i.e., 0 < ε 1, the solution of NLSW is perturbed from that of NLS with a function oscillating in time with O(ε 2 )-wavelength at O(ε 4 ) and O(ε 2 ) amplitudes for well-prepared and ill-prepared initial data, respectively. This high oscillation of the solution in time brings significant difficulties in establishing error estimates uniformly in ε of the standard finite difference methods for NLSW, such as the conservative Crank-Nicolson finite difference (CNFD) method, and the semi-implicit finite difference (SIFD) method. We obtain error bounds uniformly in ε, at the order of O(h 2 + τ ) and O(h 2 + τ 2/3 ) with time step τ and mesh size h for well-prepared and ill-prepared initial data, respectively, for both CNFD and SIFD in the l 2 -norm and discrete semi-H 1 norm. Our error bounds are valid for general nonlinearity in NLSW and for one, two, and three dimensions. To derive these uniform error bounds, we combine ε-dependent error estimates of NLSW, ε-dependent error bounds between the numerical approximate solutions of NLSW and the solution of NLS, together with error bounds between the solutions of NLSW and NLS. Other key techniques in the analysis include the energy method, cut-off of the nonlinearity, and a posterior bound of the numerical solutions by using the inverse inequality and discrete semi-H 1 norm estimate. Finally, numerical results are reported to confirm our error estimates of the numerical methods and show that the convergence rates are sharp in the respective parameter regimes.Key words. nonlinear Schrödinger equation with wave operator, energy method, error estimates, conservative Crank-Nicolson finite difference method, semi-implicit finite difference method

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Section: Lemma 34 Under the Assumptions In Theoremsupporting

confidence: 64%

Section: Finite Difference Schemes and Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Uniform Error Estimates of Finite Difference Methods for the Nonlinear Schrödinger Equation with Wave Operator

Bao¹,

Cai²

2012

SIAM J. Numer. Anal.

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156

117

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“…To test the performance of our method, we compare it with some representative methods in the literature for computing the dynamics of rotating BECs, including the Crank-Nicolson finite difference (CNFD) method [9,29,38], the semi-implicit finite difference (SIFD) method [9,29], and the TSADI method [14]. Note that these methods solve the GPE in Eulerian coordinates.…”

Section: Comparisons Of Different Methodsmentioning

confidence: 99%

“…In [13], a generalized Laguerre-FourierHermite pseudospectral method was presented in polar/cylindrical coordinates, which has spectral order of accuracy in all spatial directions. These methods have higher spatial accuracy compared to the finite difference/element methods in [3,9,29] and are also valid in dissipative variants of the GPE (1.1); cf. [46].…”

mentioning

confidence: 99%

A Simple and Efficient Numerical Method for Computing the Dynamics of Rotating Bose--Einstein Condensates via Rotating Lagrangian Coordinates

Bao¹,

Marahrens²,

Tang³

et al. 2013

SIAM J. Sci. Comput.

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Abstract. We propose a simple, efficient, and accurate numerical method for simulating the dynamics of rotating Bose-Einstein condensates (BECs) in a rotational frame with or without longrange dipole-dipole interaction (DDI). We begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with an angular momentum rotation term and/or long-range DDI, state the twodimensional (2D) GPE obtained from the 3D GPE via dimension reduction under anisotropic external potential, and review some dynamical laws related to the 2D and 3D GPEs. By introducing a rotating Lagrangian coordinate system, the original GPEs are reformulated to GPEs without the angular momentum rotation, which is replaced by a time-dependent potential in the new coordinate system. We then cast the conserved quantities and dynamical laws in the new rotating Lagrangian coordinates. Based on the new formulation of the GPE for rotating BECs in the rotating Lagrangian coordinates, a time-splitting spectral method is presented for computing the dynamics of rotating BECs. The new numerical method is explicit, simple to implement, unconditionally stable, and very efficient in computation. It is spectral-order accurate in space and second-order accurate in time and conserves the mass on the discrete level. We compare our method with some representative methods in the literature to demonstrate its efficiency and accuracy. In addition, the numerical method is applied to test the dynamical laws of rotating BECs such as the dynamics of condensate width, angular momentum expectation, and center of mass, and to investigate numerically the dynamics and interaction of quantized vortex lattices in rotating BECs without or with the long-range DDI. 1. Introduction. Bose-Einstein condensation (BEC), first observed in 1995 [4,18,23], has provided a platform to study the macroscopic quantum world. Later, with the observation of quantized vortices [2,19,34,35,37,39,50], rotating BECs have been extensively studied in the laboratory. The occurrence of quantized vortices is a hallmark of the superfluid nature of BECs. In addition, condensation of bosonic atoms and molecules with significant dipole moments whose interaction is both nonlocal and anisotropic has recently been achieved experimentally in trapped 52 Cr and 164 Dy gases [1,22,27,32,33,36,48].At temperatures T much smaller than the critical temperature T c , the properties of BEC in a rotating frame with long-range dipole-dipole interaction (DDI) are well

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Numerical study of Sivashinsky equation using a splitting scheme based on Crank‐Nicolson method

Abazari

Yildirim

2019

Math Methods in App Sciences

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The mathematical modeling of a planar solid‐liquid interface in the solidification of a dilute binary alloy is formulating by one of nonintegrable, nonlinear evolution equation known as Sivashinsky equation. In the first part of this paper, the mathematical modeling of Sivashinsky equation is briefly discussed. Since, the exact solutions of this equation is yet unknown, obtaining its numerical solution plays an important role to simulate its behavior. Therefore, in the second part, a second‐order splitting finite difference scheme, based on Crank‐Nicolson method, is investigated to approximate the solution of the Sivashinsky equation with homogeneous boundary conditions. We prove the solvability of the present scheme and establish the error estimate of the numerical scheme.

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

Cited by 155 publications

References 43 publications

Uniform Error Estimates of Finite Difference Methods for the Nonlinear Schrödinger Equation with Wave Operator

Uniform Error Estimates of Finite Difference Methods for the Nonlinear Schrödinger Equation with Wave Operator

A Simple and Efficient Numerical Method for Computing the Dynamics of Rotating Bose--Einstein Condensates via Rotating Lagrangian Coordinates

Numerical study of Sivashinsky equation using a splitting scheme based on Crank‐Nicolson method

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