Efficient Numerical Schemes for the Regularized Logarithmic Schrödinger Equation with Dirac Delta Potential

Temel Kavramlar

The authors propose and analyze efficient conservative Crank-Nicolson-type finite difference schemes for the regularized logarithmic Schrödinger equation with Dirac delta potential in 1D, establishing optimal error estimates and demonstrating the accuracy and efficiency of the numerical method through numerical experiments.

Özet

The content discusses the numerical solution of the regularized logarithmic Schrödinger equation (RLSE) with a Dirac delta potential in 1D. The key points are:

1. 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.

2. 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.

3. 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.

4. Numerical experiments are provided to support the analysis and demonstrate the accuracy and efficiency of the CNFD scheme.

5. 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.

İstatistikler

The following sentences contain key metrics or important figures:
The mass, momentum, and "regularized" energy are formally conserved for the RLSE.
The CNFD scheme satisfies the conservation laws: the mass is conserved, and the energy is conserved up to the order of the local truncation error.
The CNFD scheme achieves second-order convergence in both time and space under appropriate assumptions.

Alıntılar

"The regularized logarithmic Schrödinger equation with a small regularized parameter 0 < ε ≪ 1 is adopted to approximate the logarithmic Schrödinger equation (LSE) with linear convergence rate O(ε)."
"The Crank-Nicolson finite difference methods enjoy the second-order convergence rate in time and space."

Önemli Bilgiler Şuradan Elde Edildi

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

by Xuanxuan Zho... : arxiv.org 04-25-2024

https://arxiv.org/pdf/2404.15791.pdf

Daha Derin Sorular

How can the proposed numerical scheme be extended to higher dimensional RLSE with Dirac delta potential

To extend the proposed numerical scheme to higher dimensional RLSE with Dirac delta potential, we can utilize a similar domain decomposition technique as in the 1D case. By dividing the higher-dimensional domain into subdomains and applying appropriate interface conditions, we can construct a finite difference scheme that accounts for the Dirac delta potential in each dimension. The discretization would involve solving the RLSE equation in each subdomain while ensuring consistency and conservation across the interfaces. Additionally, the spatial mesh size and time step would need to be adjusted to accommodate the higher dimensionality of the problem.

What are the potential applications of the RLSE model beyond quantum mechanics and how can the numerical method be adapted to those applications

The RLSE model with Dirac delta potential has applications beyond quantum mechanics in various fields such as condensed matter physics, nonlinear optics, and fluid dynamics. In condensed matter physics, it can be used to study the behavior of electrons in materials with impurities or defects. In nonlinear optics, the model can describe the propagation of light in nonlinear media with localized nonlinearities. In fluid dynamics, it can be applied to study wave phenomena in fluids with localized disturbances. To adapt the numerical method for these applications, the domain decomposition technique can be tailored to the specific physical context, and the discretization parameters can be optimized based on the characteristics of the problem.

Is it possible to develop even more efficient numerical schemes for the RLSE with Dirac delta potential, perhaps by exploiting the specific structure of the problem

It is possible to develop more efficient numerical schemes for the RLSE with Dirac delta potential by exploiting the specific structure of the problem. One approach could be to incorporate adaptive mesh refinement techniques that dynamically adjust the mesh size based on the solution behavior. This can help concentrate computational resources in regions of interest, such as around the Dirac delta potential, leading to more accurate results with fewer computational resources. Additionally, utilizing higher-order finite difference schemes or spectral methods can improve the accuracy and convergence rate of the numerical solution. By carefully designing the numerical method to leverage the properties of the RLSE equation, it is possible to achieve higher efficiency and accuracy in solving the problem.