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.

翻译：本文针对一维空间中带狄拉克δ势的正则化对数薛定谔方程，提出了一种守恒型Crank-Nicolson格式有限差分方法。采用带小正则化参数$0<\eps \ll 1$的正则化对数薛定谔方程来逼近对数薛定谔方程，其线性收敛速度为$O(\eps)$。该数值方法可有效避免因对数非线性项引起的数值爆炸和/或抑制舍入误差。进而，通过区域分解技术将原始问题转化为界面问题。对界面条件的不同处理方式导出不同的离散格式，结果表明狄拉克势的简单离散近似恰好对应于其中一种守恒型有限差分格式。我们研究了有限差分格式的最优$H^1$误差估计和守恒性质。Crank-Nicolson有限差分方法在时间和空间上均具有二阶收敛速度。数值算例验证了理论分析，并展示了该数值方法的精确性和高效性。