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Calcolo
Published in: Calcolo 4/2023

01-11-2023

Accurate and efficient numerical methods for the nonlinear Schrödinger equation with Dirac delta potential

Authors: Xuanxuan Zhou, Yongyong Cai, Xingdong Tang, Guixiang Xu

Published in: Calcolo | Issue 4/2023

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Abstract

In this paper, we introduce two conservative Crank–Nicolson type finite difference schemes and a Chebyshev collocation scheme for the nonlinear Schrödinger equation with a Dirac delta potential in 1D. The key to the proposed methods is to transform the original problem into an interface problem. Different treatments on the interface conditions lead to different discrete schemes and it turns out that a simple discrete approximation of the Dirac potential coincides with one of the conservative finite difference schemes. The optimal \(H^1\) error estimates and the conservative properties of the finite difference schemes are investigated. Both Crank-Nicolson finite difference methods enjoy the second-order convergence rate in time, and the first-order/second-order convergence rates in space, depending on the approximation of the interface condition. Furthermore, the Chebyshev collocation method has been established by the domain-decomposition techniques, and it is numerically verified to be second-order convergent in time and spectrally accurate in space. Numerical examples are provided to support our analysis and study the orbital stability and the motion of the solitary solutions.
previous article Elliptic polytopes and invariant norms of linear operators
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Metadata
Title
Accurate and efficient numerical methods for the nonlinear Schrödinger equation with Dirac delta potential
Authors
Xuanxuan Zhou
Yongyong Cai
Xingdong Tang
Guixiang Xu
Publication date
01-11-2023
Publisher
Springer International Publishing
Published in
Calcolo / Issue 4/2023
Print ISSN: 0008-0624
Electronic ISSN: 1126-5434
DOI
https://doi.org/10.1007/s10092-023-00551-3

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