Error estimates of a regularized finite difference method for the Logarithmic Schr\"odinger equation with Dirac delta potential
<|reference_start|>Error estimates of a regularized finite difference method for the Logarithmic Schr\"odinger equation with Dirac delta potential: In this paper, we introduce a conservative Crank-Nicolson-type finite difference schemes for the regularized logarithmic Schr\"{o}dinger equation (RLSE) with Dirac delta potential in 1D. The regularized logarithmic Schr\"{o}dinger equation with a small regularized parameter $0<\eps \ll 1$ is adopted to approximate the logarithmic Schr\"{o}dinger equation (LSE) with linear convergence rate $O(\eps)$. The numerical method can be used to avoid numerical blow-up and/or to suppress round-off error due to the logarithmic nonlinearity in LSE. Then, by using domain-decomposition technique, we can 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. The Crank-Nicolson finite difference methods enjoy the second-order convergence rate in time and space. Numerical examples are provided to support our analysis and show the accuracy and efficiency of the numerical method.<|reference_end|>
@article{zhou2024error,
    title={Error estimates of a regularized finite difference method for the
  Logarithmic Schr\"{o}dinger equation with Dirac delta potential},
    author={Xuanxuan Zhou and Tingchun Wang and Yong Wu and Yongyong Cai},
    journal={arXiv preprint arXiv:2404.15791},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.15791},
    primaryClass={math.NA cs.NA}
}