An exponential wave integrator Fourier pseudospectral method for the nonlinear Schrödinger equation with wave operator

Ji, Bingquan; Zhang, Luming

doi:10.1007/s12190-017-1144-x

Cited by 9 publications

(6 citation statements)

References 13 publications

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“…and homogeneous boundary conditions ϕ(−40, τ) = 0 = ϕ (40, τ). This problem is challenging due to the blow up phenomena of the soliton wave and has been considered in many papers as a practical problem [1,11,12] which has no exact solution. In Figure 3 the soliton wave is presented at different T, where the propagation of soliton is clearly seen as time increases.…”

Section: Test Casesmentioning

confidence: 99%

“….5530 × 10 −3 1.5323 × 10 −3 4.8039 × 10 −3 0.000948069171383 0.000232841089347i 16 6.0002 × 10 −3 2.2329 × 10 −3 6.4022 × 10 −3 0.000985139322483 0.000254685053319i 32 6.7727 × 10 −3 2.7085 × 10 −3 7.2942 × 10 −3 0.000991057419645 0.000255807453310i 64 6.9416 × 10 −3 2.8066 × 10 −3 7.4875 × 10 −3 0.000992332262253 0.000256101948732i 128 6.9769 × 10 −3 2.8255 × 10 −3 7.5273 × 10 −3 0.000992632417626 0.000256178051069i 256 6.9852 × 10 −3 2.8299 × 10 −3 7.5367 × 10 −3 0.000992706293238 0.000256197214486i 512 6.9872 × 10 −3 2.8309 × 10 −3 7.5390 × 10 −3 0.000992724689529 0.000256202013479i Fourier pseudospectral method [12] Fourier pseudospectral method [12] Proposed HWCM Proposed HWCM…”

Section: E Rementioning

confidence: 99%

“…Due to the above mentioned applications, different numerical techniques have been developed to solve the important NHSEs. These numerical technique are finite difference method [5][6][7][8][9], spline collocation methods [10] Galerkin method [11], Fourier pseudospectral method [12] and Multi-symplectic integrator [1]. The nonconservative scheme [5] may not handle the nonlinear blow up phenomena easily and the conservative schemes are presented to minimize these issues [7,9].…”

Section: Introductionmentioning

confidence: 99%

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Applications of Haar Wavelet-Finite Difference Hybrid Method and Its Convergence for Hyperbolic Nonlinear Schrödinger Equation with Energy and Mass Conversion

et al. 2021

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This article is concerned with the numerical solution of nonlinear hyperbolic Schro¨dinger equations (NHSEs) via an efficient Haar wavelet collocation method (HWCM). The time derivative is approximated in the governing equations by the central difference scheme, while the space derivatives are replaced by finite Haar series, which transform it to full algebraic form. The experimental rate of convergence follows the theoretical statements of convergence and the conservation laws of energy and mass are also presented, which strengthens the proposed method to be convergent and conservative. The Haar wavelets based on numerical results for solitary wave shape of |φ| are discussed in detail. The proposed approach provides a fast convergent approximation to the NHSEs. The reliability and efficiency of the method are illustrated by computing the maximum error norm and the experimental rate of convergence for different problems. Comparisons are performed with various existing methods in recent literature and better performance of the proposed method is shown in various tables and figures.

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Section: Test Casesmentioning

confidence: 99%

Section: E Rementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Applications of Haar Wavelet-Finite Difference Hybrid Method and Its Convergence for Hyperbolic Nonlinear Schrödinger Equation with Energy and Mass Conversion

et al. 2021

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“…However, the proposed scheme is fully implicit. Although some explicit exponential integrators were proposed in [1,17,51], but they failed to be energy-preserving. Gu et al developed linearly implicit exponential partitioned AVF methods in [13].…”

Section: Introductionmentioning

confidence: 99%

Linearly implicit energy-preserving integrating factor methods for the 2D nonlinear Schrödinger equation with wave operator and convergence analysis

Gu¹,

Cai²,

Jiang³

et al. 2022

Preprint

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In this paper, we develop a novel class of linear energy-preserving integrating factor methods for the 2D nonlinear Schrödinger equation with wave operator (NLSW), combining the scalar auxiliary variable approach and the integrating factor methods. A second-order scheme is proposed, which is rigorously proved to be energy-preserving. By using the energy methods, we analyze its optimal convergence in the H 1 norm without any restrictions on the grid ratio, where a novel technique and an improved induction argument are proposed to overcome the difficulty posed by the unavailability of a priori L ∞ estimates of numerical solutions. Based on the integrating factor Runge-Kutta methods, we extend the proposed scheme to arbitrarily high order, which is also linear and conservative. Numerical experiments are presented to confirm the theoretical analysis and demonstrate the advantages of the proposed methods.

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“…Several numerical algorithms have been studied for solving the NSEW (Refs. [3][4][5][6][7][8][9][10][11] and references therein). Recently, structure-preserving algorithms were proposed to solving the Hamiltonian systems [12][13][14][15] and applied to various PDEs, such as the nonlinear Schrödingertype equation [16][17][18][19], wave equation [20], and KdV equation [21].…”

Section: Introductionmentioning

confidence: 99%

Compact Local Structure-Preserving Algorithms for the Nonlinear Schrödinger Equation with Wave Operator

Lang-yang

Tian

Cai

2020

Mathematical Problems in Engineering

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Combining the compact method with the structure-preserving algorithm, we propose a compact local energy-preserving scheme and a compact local momentum-preserving scheme for the nonlinear Schrödinger equation with wave operator (NSEW). The convergence rates of both schemes are Oh4+τ2. The discrete local conservative properties of the presented schemes are derived theoretically. Numerical experiments are carried out to demonstrate the convergence order and local conservation laws of the developed algorithms.

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An exponential wave integrator Fourier pseudospectral method for the nonlinear Schrödinger equation with wave operator

Cited by 9 publications

References 13 publications

Applications of Haar Wavelet-Finite Difference Hybrid Method and Its Convergence for Hyperbolic Nonlinear Schrödinger Equation with Energy and Mass Conversion

Applications of Haar Wavelet-Finite Difference Hybrid Method and Its Convergence for Hyperbolic Nonlinear Schrödinger Equation with Energy and Mass Conversion

Linearly implicit energy-preserving integrating factor methods for the 2D nonlinear Schrödinger equation with wave operator and convergence analysis

Compact Local Structure-Preserving Algorithms for the Nonlinear Schrödinger Equation with Wave Operator

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