The content discusses the numerical solution of the regularized logarithmic Schrödinger equation (RLSE) with a Dirac delta potential in 1D. The key points are:
The RLSE is introduced to approximate the logarithmic Schrödinger equation (LSE) with Dirac delta potential, which has difficulties due to the singularity of the logarithmic nonlinearity.
The authors propose a conservative Crank-Nicolson-type finite difference scheme (CNFD) for the RLSE, which can be reformulated as a simple discrete approximation of the Dirac delta potential.
The authors prove the optimal H1 error estimates and the conservative properties of the CNFD scheme, showing it enjoys second-order convergence in both time and space.
Numerical experiments are provided to support the analysis and demonstrate the accuracy and efficiency of the CNFD scheme.
The authors use domain decomposition techniques to transform the original problem into an interface problem, leading to different discrete schemes with the simple discrete approximation of the Dirac delta potential coinciding with one of the conservative finite difference schemes.
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by Xuanxuan Zho... at arxiv.org 04-25-2024
https://arxiv.org/pdf/2404.15791.pdfDeeper Inquiries