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Low regularity error estimates for the time integration of 2D NLS

Lun Ji, Alexander Ostermann, Frédéric Rousset, Katharina Schratz

TL;DR

This work develops and analyzes a filtered Lie splitting method for the 2D cubic nonlinear Schrödinger equation on the torus, achieving convergence for rough data with $s>0$ by leveraging discrete Bourgain spaces. The core idea is to couple a frequency-filtered splitting scheme with sharp discrete multilinear estimates, enabling an $L^2$-error rate of $\tau^{s/2}$ without requiring smooth initial data. The authors establish local and global error bounds, prove boundedness of the discrete solution in the appropriate Bourgain norms, and provide a comprehensive proof of the discrete key nonlinear estimates. Numerical experiments corroborate the sharpness of the rate and demonstrate the method’s effectiveness for rough initial data, highlighting the scheme’s potential for efficient simulations in low-regularity regimes.

Abstract

A filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schrödinger equation on the two-dimensional torus $\mathbb{T}^2$. The scheme is analyzed in a framework of discrete Bourgain spaces, which allows us to consider initial data with low regularity; more precisely initial data in $H^s(\mathbb{T}^2)$ with $s>0$. In this way, the usual stability restriction to smooth Sobolev spaces with index $s>1$ is overcome. Rates of convergence of order $τ^{s/2}$ in $L^2(\mathbb{T}^2)$ at this regularity level are proved. Numerical examples illustrate that these convergence results are sharp.

Low regularity error estimates for the time integration of 2D NLS

TL;DR

This work develops and analyzes a filtered Lie splitting method for the 2D cubic nonlinear Schrödinger equation on the torus, achieving convergence for rough data with by leveraging discrete Bourgain spaces. The core idea is to couple a frequency-filtered splitting scheme with sharp discrete multilinear estimates, enabling an -error rate of without requiring smooth initial data. The authors establish local and global error bounds, prove boundedness of the discrete solution in the appropriate Bourgain norms, and provide a comprehensive proof of the discrete key nonlinear estimates. Numerical experiments corroborate the sharpness of the rate and demonstrate the method’s effectiveness for rough initial data, highlighting the scheme’s potential for efficient simulations in low-regularity regimes.

Abstract

A filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schrödinger equation on the two-dimensional torus . The scheme is analyzed in a framework of discrete Bourgain spaces, which allows us to consider initial data with low regularity; more precisely initial data in with . In this way, the usual stability restriction to smooth Sobolev spaces with index is overcome. Rates of convergence of order in at this regularity level are proved. Numerical examples illustrate that these convergence results are sharp.
Paper Structure (9 sections, 18 theorems, 221 equations, 2 figures)

This paper contains 9 sections, 18 theorems, 221 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Analysis of the exact solution
  3. Discrete Bourgain spaces
  4. Boundedness of the exact solution in discrete Bourgain spaces
  5. Local error estimates
  6. Global error estimates
  7. Proof of Theorem \ref{['mainthm']}
  8. Proof of Theorem \ref{['disckeythm']}
  9. Numerical experiments

Key Result

Theorem 1.1

Let $u_0\in H^{s_0}(\mathbb{T}^2),~s_0\in (0, 2]$ and $T>0$ such that there exists an exact solution $u$ of n with initial data $u_0$ such that $u\in X^{s_0, b_0}(T)\subset\mathcal{C}([0,T],H^{s_0})$. Let $u_n$ be the numerical solution defined by the scheme 0. Then, we have the following error esti

Figures (2)

  • Figure 1: $L^2$ error of the filtered Lie splitting scheme for rough initial data $u_0\in H^s$. (a) $s=0.2$; (b) $s={1/3}$; (c) $s={0.5}$; (d) $s=1$.
  • Figure 2: $L^2$ error of the filtered Lie splitting scheme for rough initial data $u_0\in H^{0.1}$ with different reference solutions. (a) Reference solution with largest Fourier mode $K=(2^{10},2^{10})$, spatial mesh size $\Delta x=0.0061$, and time step size $\tau=2^{-18}$; (b) Reference solution with largest Fourier mode $K=(2^{12}, 2^{12})$, spatial mesh size $\Delta x=0.0015$, and time step size $\tau=2^{-22}$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Lemma 2.1
  • Proposition 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Lemma 2.5
  • Corollary 2.6
  • proof
  • ...and 21 more