Analysis of some new conservative schemes for nonlinear Schrödinger equation with wave operator

Wang, Tingchun; Zhang, Luming

doi:10.1016/j.amc.2006.06.015

Cited by 64 publications

(40 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Error bounds of conservative Crank-Nicolson finite difference (CNFD) for NLS in one dimension (1D) have been established in [9,13]. For NLSW in 1D with ε = O(1), the error estimates of conservative finite difference schemes have been obtained in [26]. However, the proofs in [26] rely strongly on the conservative properties of the schemes and the discrete version of the Sobolev inequality in 1D,…”

Section: Introductionmentioning

confidence: 99%

“…For NLSW in 1D with ε = O(1), the error estimates of conservative finite difference schemes have been obtained in [26]. However, the proofs in [26] rely strongly on the conservative properties of the schemes and the discrete version of the Sobolev inequality in 1D,…”

Section: Introductionmentioning

confidence: 99%

“…However, few numerical methods have been considered for NLSW in the literature, and most of them are the conservative finite difference methods [10,14,26]. For the corresponding error analysis on the split error for NLS, see [8,11,17,19] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…Follow the analogous arguments of the CNFD method for NLS [9,13] and NLSW [14,26]. We omit the details here for brevity.…”

mentioning

confidence: 99%

See 3 more Smart Citations

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

Bao¹,

Cai²

2012

SIAM J. Numer. Anal.

164

117

View full text Add to dashboard Cite

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

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Follow the analogous arguments of the CNFD method for NLS [9,13] and NLSW [14,26]. We omit the details here for brevity.…”

mentioning

confidence: 99%

See 2 more Smart Citations

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

Bao¹,

Cai²

2012

SIAM J. Numer. Anal.

164

117

View full text Add to dashboard Cite

show abstract

“…In [24], Zhang and Chang proposed a four-level explicit and conservative scheme. Wang and Zhang [25] developed some different conservative schemes based on some special techniques on the nonlinear terms. Hu and Chan [26] further considered a conservative difference scheme for two-dimensional NLSE.…”

Section: Introductionmentioning

confidence: 99%

Several conservative compact schemes for a class of nonlinear Schrödinger equations with wave operator

Cheng

2018

Bound Value Probl

View full text Add to dashboard Cite

In this paper, several different conserving compact finite difference schemes are developed for solving a class of nonlinear Schrödinger equations with wave operator. It is proved that the numerical solutions are bounded and the numerical methods can achieve a convergence rate of O(τ 2 + h 4 ) in the maximum norm. Moreover, by applying Richardson extrapolation, the proposed methods have a convergence rate of O(τ 4 + h 4 ) in the maximum norm. Finally, several numerical experiments are presented to illustrate the theoretical results. MSC: 65N06; 65N12

show abstract

Local absorbing boundary conditions for two‐dimensional nonlinear Schrödinger equation with wave operator on unbounded domain

2021

Math Methods in App Sciences

View full text Add to dashboard Cite

The paper is devoted to designing the local absorbing boundary conditions for nonlinear Schrödinger equation with wave operator on unbounded domain in two dimensions. Introduce the artificial boundaries and find the appropriate absorbing boundary conditions for the original problem, which lead to a bounded computational domain. Constructing the local absorbing boundary conditions on the artificial boundaries, the original problem defined on unbounded domain is reduced to an initial boundary value problem on the bounded computational domain, which can be efficiently solved by the finite difference method. The stability of the reduced problem with the local absorbing boundary conditions is rigorously analyzed by introducing a series of auxiliary variables. The numerical results illustrate that the proposed approach is effective and feasible.

show abstract

Analysis of some new conservative schemes for nonlinear Schrödinger equation with wave operator

Cited by 64 publications

References 10 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

Several conservative compact schemes for a class of nonlinear Schrödinger equations with wave operator

Local absorbing boundary conditions for two‐dimensional nonlinear Schrödinger equation with wave operator on unbounded domain

Contact Info

Product

Resources

About