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PreprintCC BY 4.0arXiv

Unconditional optimal convergence of a relaxation Crank-Nicolson difference scheme for 2D space-fractional NLS

  • Lisen Ding
  • Dongling Wang
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Abstract

In this paper, we develop a linearly implicit relaxation Crank-Nicolson difference scheme for the two-dimensional space-fractional nonlinear Schrödinger (SFNS) equation, where the fractional Laplacian is approximated by a simple fractional central difference scheme. The proposed scheme is uniquely solvable and conserves discrete mass for general real-valued nonlinear coefficients, and a modified discrete energy in the cubic case. By introducing a modified error system employing a cut-off technique, we establish the optimal-order error estimates without mesh-ratio restriction. Specifically, the primary variable achieves second-order temporal and spatial convergence in the discrete Hhα/2-norm, and the relaxation variable satisfies the same convergence in the discrete L2-norm. Some numerical experiments confirm the relevant theoretical results.

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Cite

BibTeXRIS
BibTeX
@misc{ding2026unconditional,
  title = {{Unconditional optimal convergence of a relaxation Crank-Nicolson difference scheme for 2D space-fractional NLS}},
  author = {Lisen Ding and Dongling Wang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00571},
  eprint = {2611.00571},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00571},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Unconditional optimal convergence of a relaxation Crank-Nicolson difference scheme for 2D space-fractional NLS
AU  - Lisen Ding
AU  - Dongling Wang
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00571
UR  - https://arxiv.org/abs/2611.00571
AB  - In this paper, we develop a linearly implicit relaxation Crank-Nicolson difference scheme for the two-dimensional space-fractional nonlinear Schrödinger (SFNS) equation, where the fractional Laplacian is approximated by a simple fractional central difference scheme. The proposed scheme is uniquely solvable and conserves discrete mass for general real-valued nonlinear coefficients, and a modified discrete energy in the cubic case. By introducing a modified error system employing a cut-off technique, we establish the optimal-order error estimates without mesh-ratio restriction. Specifically, the primary variable achieves second-order temporal and spatial convergence in the discrete H_h^α/2-norm, and the relaxation variable satisfies the same convergence in the discrete L^2-norm. Some numerical experiments confirm the relevant theoretical results.
ER  -