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Low regularity full error estimates for the cubic nonlinear Schrödinger equation

Lun Ji, Alexander Ostermann, Frédéric Rousset, Katharina Schratz

TL;DR

The paper tackles the challenge of accurately solving the cubic nonlinear Schrödinger equation on the 2D torus with low-regularity data. It combines a pseudospectral spatial discretization with a filtered Lie splitting in time and analyzes the scheme within discrete Bourgain spaces to handle rough initial data. The authors prove an $L^2$-convergence rate of $O(\tau^{s/2}+N^{-s})$ for data in $H^s(\mathbb{T}^2)$, $s>0$, and establish a global error bound that accounts for both time stepping and spectral truncation. Numerical experiments corroborate the theory, showing the predicted convergence rates across a range of $s$ values and highlighting the method’s robustness for low-regularity inputs.

Abstract

For the numerical solution of the cubic nonlinear Schrödinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting scheme in time is considered. This scheme is shown to converge even for initial data with very low regularity. In particular, for data in $H^s(\mathbb T^2)$, where $s>0$, convergence of order $\mathcal O(τ^{s/2}+N^{-s})$ is proved in $L^2$. Here $τ$ denotes the time step size and $N$ the number of Fourier modes considered. The proof of this result is carried out in an abstract framework of discrete Bourgain spaces, the final convergence result, however, is given in $L^2$. The stated convergence behavior is illustrated by several numerical examples.

Low regularity full error estimates for the cubic nonlinear Schrödinger equation

TL;DR

The paper tackles the challenge of accurately solving the cubic nonlinear Schrödinger equation on the 2D torus with low-regularity data. It combines a pseudospectral spatial discretization with a filtered Lie splitting in time and analyzes the scheme within discrete Bourgain spaces to handle rough initial data. The authors prove an -convergence rate of for data in , , and establish a global error bound that accounts for both time stepping and spectral truncation. Numerical experiments corroborate the theory, showing the predicted convergence rates across a range of values and highlighting the method’s robustness for low-regularity inputs.

Abstract

For the numerical solution of the cubic nonlinear Schrödinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting scheme in time is considered. This scheme is shown to converge even for initial data with very low regularity. In particular, for data in , where , convergence of order is proved in . Here denotes the time step size and the number of Fourier modes considered. The proof of this result is carried out in an abstract framework of discrete Bourgain spaces, the final convergence result, however, is given in . The stated convergence behavior is illustrated by several numerical examples.
Paper Structure (7 sections, 10 theorems, 98 equations, 2 figures)

This paper contains 7 sections, 10 theorems, 98 equations, 2 figures.

Table of Contents

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

Key Result

Theorem 1.2

For $s_0\in(0,2]$ and initial data $u_0\in H^{s_0}(\mathbb{T}^2)$, let $u$ be the exact solution of nls in the Bourgain space $X^{s_0,b_0}(T)$ with initial data $u_0$ for appropriate $T>0$ and $b_0>\tfrac{1}{2}$ (see also Ji). Further, let $u_n$ denote the numerical solution defined by the scheme li where the constant $C_T$ depends on $\tau_0$, $N_0$ and $T$, but is independent of $n$, $\tau$ and

Figures (2)

  • Figure 1: $L^2$ error of the fully discrete Lie splitting scheme \ref{['liespace']} for rough initial data $u_0\in H^s(\mathbb{T}^2)$. We took the choice $\theta=\tau=4N^{-2}$. (a) $s=0.2$; (b) $s=1/3$; (c) $s=0.5$; (d) $s=1$.
  • Figure 2: $L^2$ error of the fully discrete Lie splitting scheme for rough initial data $u_0\in H^{0.1}$ with different reference solutions. We took the choice $\theta=\tau=4N^{-2}$. (a) Reference solution with largest Fourier mode $K=(2^{12},2^{12})$, spatial mesh size $\Delta x=0.0015$, and time step size $\tau=2^{-22}$; (b) Reference solution with largest Fourier mode $K=(2^{13}, 2^{13})$, spatial mesh size $\Delta x=0.0008$, and time step size $\tau=2^{-24}$.

Theorems & Definitions (16)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1
  • Theorem 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 6 more