Optimal error bounds on an exponential wave integrator Fourier spectral method for the logarithmic Schrödinger equation (2412.16902v1)

Abstract: We prove a nearly optimal error bound on the exponential wave integrator Fourier spectral (EWI-FS) method for the logarithmic Schr\"odinger equation (LogSE) under the assumption of $H^2$-solution, which is theoretically guaranteed. Subject to a CFL-type time step size restriction $\tau |\ln \tau| \leq h^2/|\ln h|$ for obtaining the stability of the numerical scheme affected by the singularity of the logarithmic nonlinearity, an $L^2$-norm error bound of order $O(\tau |\ln \tau|^2 + h^2 |\ln h|)$ is established, where $\tau$ is the time step size and $h$ is the mesh size. Compared to the error estimates of the LogSE in the literature, our error bound either greatly improves the convergence rate under the same regularity assumptions or significantly weakens the regularity requirement to obtain the same convergence rate. Moreover, our result can be directly applied to the LogSE with low regularity $L^\infty$-potential, which is not allowed in the existing error estimates. Two main ingredients are adopted in the proof: (i) an $H^2$-conditional $L^2$-stability estimate, which is established using the energy method to avoid singularity of the logarithmic nonlinearity, and (ii) mathematical induction with inverse inequalities to control the $H^2$-norm of the numerical solution. Numerical results are reported to confirm our error estimates and demonstrate the necessity of the time step size restriction imposed. We also apply the EWI-FS method to investigate soliton collisions in one dimension and vortex dipole dynamics in two dimensions.

• The paper introduces the EWI-FS method that integrates an exponential wave integrator with a Fourier spectral approach to address the logarithmic nonlinearity.
• It derives nearly optimal L2-error estimates of O(τ|lnτ|² + h²|lnh|) under a CFL-type restriction to ensure numerical accuracy and stability.
• The method is rigorously validated through theoretical analysis and numerical experiments, establishing a robust framework for simulating nonlinear wave dynamics.

Analysis of an Exponential Wave Integrator Fourier Spectral Method for the Logarithmic Schrödinger Equation

The paper focuses on developing a robust numerical method for the Logarithmic Schrödinger Equation (LogSE), which is pivotal in nonlinear wave mechanics and several applications, such as quantum mechanics, optics, and Bose-Einstein condensates. This equation presents unique challenges for numerical methods due to its logarithmic nonlinearity inherently leading to singular behavior. The authors propose an Exponential Wave Integrator Fourier Spectral (EWI-FS) method, providing explicit error bounds, crucial for validating the reliability and efficiency of the method under low regularity potential.

Key Contributions and Methodology

1. Formulation of the EWI-FS Method: The authors devise a first-order exponential wave integrator in combination with a Fourier spectral method to discretize the LogSE. This approach effectively handles the propagative nature of the Schrödinger equation while ensuring spectral accuracy in space.
2. Optimal Error Bounds: Under the assumption of $H^2$-solutions, the paper derives nearly optimal error bounds of $O(\tau |\ln \tau|^2 + h^2 |\ln h|)$ in the $L^2$-norm. These are achieved subject to a CFL-type restriction $\tau |\ln \tau| \leq h^2/|\ln h|$, which compensates for the singularity due to the logarithmic nonlinearity. The method shows robustness in numerical stability and provides improved convergence rates compared to existing techniques.
3. Theoretical and Numerical Validation: Through rigorous error analysis and numerical experiments, the method demonstrates superior convergence behavior. Notably, the derived CFL-type time-step condition is empirically verified as essential, highlighting the importance of proper discretization in handling nonlinear singularities.

Analytical Techniques and Propositions

• The stability analysis is pivotal, employing a conditional $L^2$-stability estimate that takes advantage of properties unique to the logarithmic term. This involves using induction and inverse inequalities to manage the $H^2$ norm of the numerical solution.
• Local truncation errors are systematically controlled by judiciously selecting parameters based on the logarithmic nature, showing that $L^2$-error grows sublinearly with temporal step size and mesh refinement when adhering to the CFL condition.

Implications and Future Work

The proposal of an EWI-FS method signifies a meaningful step towards handling nonlinear dispersive equations with singular nonlinear terms. Theoretical advancements ensure practical applications in simulating complex wave phenomena, such as soliton collisions in random media and vortex dynamics.

Future research avenues could explore the extension of these techniques to higher dimensions and other forms of nonlinearities or disorders. Furthermore, refining these methods could lead to enhanced simulations of open quantum systems where the LogSE model is relevant, potentially advancing computational methodologies in quantum mechanics.

Conclusion

The paper's contribution lies not only in its immediate application to numerical simulations but also in setting a foundation for future research into associated nonlinear problems. The careful derivation of error estimates amid the challenges posed by LogSE's inherent singularities underscores the potential for similar strategies in broader contexts. Researchers in numerical analysis and computational physics particularly stand to gain from these insights into methodological precision for dispersive PDEs with low regularity.