Volume 25, Issue 1
Efficient Numerical Computation of Time-Fractional Nonlinear Schrödinger Equations in Unbounded Domain

Jiwei Zhang, Dongfang Li & Xavier Antoine

Commun. Comput. Phys., 25 (2019), pp. 218-243.

Published online: 2018-09

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  • Abstract

The aim of this paper is to derive a stable and efficient scheme for solving the one-dimensional time-fractional nonlinear Schrödinger equation set in an unbounded domain. We first derive absorbing boundary conditions for the fractional system by using the unified approach introduced in [47, 48] and a linearization procedure. Then, the initial boundary-value problem for the fractional system with ABCs is discretized, a stability analysis is developed and the error estimate O(h2+τ) is stated. To accelerate the L1-scheme in time, a sum-of-exponentials approximation is introduced to speed-up the evaluation of the Caputo fractional derivative. The resulting algorithm is highly efficient for long time simulations. Finally, we end the paper by reporting some numerical simulations to validate the properties (accuracy and efficiency) of the derived scheme.

  • Keywords

Time-fractional nonlinear Schr ¨odinger equation absorbing boundary condition stability analysis convergence analysis sum-of-exponentials approximation.

  • AMS Subject Headings

35R11 35Q41 65L20

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COPYRIGHT: © Global Science Press

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@Article{CiCP-25-218, author = {Jiwei Zhang, Dongfang Li and Xavier Antoine}, title = {Efficient Numerical Computation of Time-Fractional Nonlinear Schrödinger Equations in Unbounded Domain}, journal = {Communications in Computational Physics}, year = {2018}, volume = {25}, number = {1}, pages = {218--243}, abstract = {

The aim of this paper is to derive a stable and efficient scheme for solving the one-dimensional time-fractional nonlinear Schrödinger equation set in an unbounded domain. We first derive absorbing boundary conditions for the fractional system by using the unified approach introduced in [47, 48] and a linearization procedure. Then, the initial boundary-value problem for the fractional system with ABCs is discretized, a stability analysis is developed and the error estimate O(h2+τ) is stated. To accelerate the L1-scheme in time, a sum-of-exponentials approximation is introduced to speed-up the evaluation of the Caputo fractional derivative. The resulting algorithm is highly efficient for long time simulations. Finally, we end the paper by reporting some numerical simulations to validate the properties (accuracy and efficiency) of the derived scheme.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2017-0195}, url = {http://global-sci.org/intro/article_detail/cicp/12669.html} }
TY - JOUR T1 - Efficient Numerical Computation of Time-Fractional Nonlinear Schrödinger Equations in Unbounded Domain AU - Jiwei Zhang, Dongfang Li & Xavier Antoine JO - Communications in Computational Physics VL - 1 SP - 218 EP - 243 PY - 2018 DA - 2018/09 SN - 25 DO - http://dor.org/10.4208/cicp.OA-2017-0195 UR - https://global-sci.org/intro/cicp/12669.html KW - Time-fractional nonlinear Schr ¨odinger equation KW - absorbing boundary condition KW - stability analysis KW - convergence analysis KW - sum-of-exponentials approximation. AB -

The aim of this paper is to derive a stable and efficient scheme for solving the one-dimensional time-fractional nonlinear Schrödinger equation set in an unbounded domain. We first derive absorbing boundary conditions for the fractional system by using the unified approach introduced in [47, 48] and a linearization procedure. Then, the initial boundary-value problem for the fractional system with ABCs is discretized, a stability analysis is developed and the error estimate O(h2+τ) is stated. To accelerate the L1-scheme in time, a sum-of-exponentials approximation is introduced to speed-up the evaluation of the Caputo fractional derivative. The resulting algorithm is highly efficient for long time simulations. Finally, we end the paper by reporting some numerical simulations to validate the properties (accuracy and efficiency) of the derived scheme.

Jiwei Zhang, Dongfang Li & Xavier Antoine. (1970). Efficient Numerical Computation of Time-Fractional Nonlinear Schrödinger Equations in Unbounded Domain. Communications in Computational Physics. 25 (1). 218-243. doi:10.4208/cicp.OA-2017-0195
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