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Error estimates of a regularized finite difference method for the Logarithmic Schrödinger equation with Dirac delta potential

Xuanxuan Zhou, Tingchun Wang, Yong Wu, Yongyong Cai

TL;DR

This paper develops a conservative Crank–Nicolson finite difference method for the 1D regularized logarithmic Schrödinger equation with a Dirac delta potential by reformulating the problem as an interface (domain-decomposition) problem. A delta-consistent, two-subdomain CNFD scheme is designed to be mass- and energy-conserving, with a discrete delta approximation arising naturally from the interface treatment. Rigorous $H^1$ error estimates are established via an energy argument, and the method exhibits second-order convergence in space and time; numerical experiments confirm the theory and show the RLSE converges to the original LSE as the regularization parameter $\varepsilon$ tends to zero. Stability studies of standing waves align with known orbital stability results, underscoring the method’s reliability for singular nonlinear interactions.

Abstract

In this paper, we introduce a conservative Crank-Nicolson-type finite difference schemes for the regularized logarithmic Schrödinger equation (RLSE) with Dirac delta potential in 1D. The regularized logarithmic Schrödinger equation with a small regularized parameter $0<\eps \ll 1$ is adopted to approximate the logarithmic Schrödinger 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.

Error estimates of a regularized finite difference method for the Logarithmic Schrödinger equation with Dirac delta potential

TL;DR

This paper develops a conservative Crank–Nicolson finite difference method for the 1D regularized logarithmic Schrödinger equation with a Dirac delta potential by reformulating the problem as an interface (domain-decomposition) problem. A delta-consistent, two-subdomain CNFD scheme is designed to be mass- and energy-conserving, with a discrete delta approximation arising naturally from the interface treatment. Rigorous error estimates are established via an energy argument, and the method exhibits second-order convergence in space and time; numerical experiments confirm the theory and show the RLSE converges to the original LSE as the regularization parameter tends to zero. Stability studies of standing waves align with known orbital stability results, underscoring the method’s reliability for singular nonlinear interactions.

Abstract

In this paper, we introduce a conservative Crank-Nicolson-type finite difference schemes for the regularized logarithmic Schrödinger equation (RLSE) with Dirac delta potential in 1D. The regularized logarithmic Schrödinger equation with a small regularized parameter is adopted to approximate the logarithmic Schrödinger equation (LSE) with linear convergence rate . 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 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.
Paper Structure (13 sections, 6 theorems, 63 equations, 4 figures, 6 tables)

This paper contains 13 sections, 6 theorems, 63 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Finite difference schemes
  3. Error estimates
  4. Error estimates of CNFD
  5. Numerical experiments
  6. Spatial/temporal resolution
  7. Conservation laws
  8. Convergence rate of the regularized model
  9. Stability test
  10. Conclusions
  11. The a priori $l^\infty$ bounds of the numerical solutions of CNFD:
  12. Local error bound of $R_M^k$ of CNFD:
  13. The bound of $\hat{Q}_j^k$ and $\delta_x^+\hat{Q}_j^k$:

Key Result

lemma 1

Caizhou:2023 For any mesh functions $U,V\in Z_h^0$, there hold

Figures (4)

  • Figure 1: The conservation laws of CNFD with initial value \ref{['ini:1']} for different $\varepsilon$.
  • Figure 2: Convergence rate of the regularized model for different $\epsilon$.
  • Figure 3: The log-plot of $e(\omega,\eta)$ and $e_{\theta}(\omega,\eta)$ for different $\eta$ of the perturbation \ref{['perturbation_2']} under $\lambda=-1,T=30$.
  • Figure 4: The plot of $e(\omega,\eta)$ and $e_{\theta}(\omega,\eta)$ for different $\eta$ of the perturbation \ref{['perturbation_2']} under $\lambda=1,T=50$.

Theorems & Definitions (13)

  • Remark 2.1
  • lemma 1
  • lemma 2
  • proof
  • Remark 2.2
  • Theorem 3.1
  • lemma 3
  • proof
  • lemma 4
  • proof
  • ...and 3 more