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Low regularity error estimates for high dimensional nonlinear Schrödinger equations

Lun Ji, Alexander Ostermann

Abstract

The filtered Lie splitting scheme is an established method for the numerical integration of the periodic nonlinear Schrödinger equation at low regularity. Its temporal convergence was recently analyzed in a framework of discrete Bourgain spaces in one and two space dimensions for initial data in $H^s$ with $0<s\leq 2$. Here, this analysis is extended to dimensions $d=3, 4, 5$ for data satisfying $d/2-1 < s \leq 2$. In this setting, convergence of order $s/2$ in $L^2$ is proven. Numerical examples illustrate these convergence results.

Low regularity error estimates for high dimensional nonlinear Schrödinger equations

Abstract

The filtered Lie splitting scheme is an established method for the numerical integration of the periodic nonlinear Schrödinger equation at low regularity. Its temporal convergence was recently analyzed in a framework of discrete Bourgain spaces in one and two space dimensions for initial data in with . Here, this analysis is extended to dimensions for data satisfying . In this setting, convergence of order in is proven. Numerical examples illustrate these convergence results.
Paper Structure (5 sections, 7 theorems, 80 equations, 2 figures, 1 table)

This paper contains 5 sections, 7 theorems, 80 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. A Bourgain space framework
  3. Local error estimates
  4. Proof of Theorem \ref{['mainthm']}
  5. Numerical experiments

Key Result

Theorem 1.1

For given $1\leq d\leq5$ and $T>0$, let $u_0\in H^{s_0}(\mathbb{T}^d)$ be such that the exact solution $u$ of nls with initial data $u_0$ exists up to time $T$ for $s_0$ as in s0b0 satisfying $s_0\le 2$. Let $u_n$ be the numerical solution defined by the scheme lie. Then, we have the following error

Figures (2)

  • Figure 1: The $L^2$ error of the filtered Lie splitting scheme for the three dimensional NLS with rough initial data $u_0\in H^s(\mathbb{T}^3)$. (a) $s=0.5$; (b) $s=0.75$; (c) $s=1$; (d) $s=2$.
  • Figure 2: The $L^2$ error of the filtered Lie splitting scheme for the four dimensional NLS with rough initial data $u_0\in H^s(\mathbb{T}^4)$. (a) $s=1$; (b) $s=2$.

Theorems & Definitions (14)

  • Theorem 1.1
  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 3.1
  • proof
  • ...and 4 more