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Improved error estimates for low-regularity integrators using space-time bounds

Maximilian Ruff

TL;DR

This work establishes optimal convergence rates for low-regularity time integrators applied to the 1D periodic nonlinear Schrödinger and wave equations with initial data in $H^1$. By deriving local-error representations and exploiting continuous-time space-time bounds—specifically the $L^4([0,T]\times \mathbb{T})$ Strichartz estimate for Schrödinger and a novel null-form bound for the wave equation—the authors prove first-order convergence for the Ostermann–Schratz exponential-type integrator and second-order convergence for the corrected Lie splitting. These results surpass previously known fractional rates in this setting and rely on continuous-time estimates to avoid discretization-induced losses. The framework is flexible and may extend to higher dimensions or fully discrete schemes, highlighting the potential of space-time analysis in numerical treatment of dispersive and hyperbolic equations.

Abstract

We prove optimal convergence rates for certain low-regularity integrators applied to the one-dimensional periodic nonlinear Schrödinger and wave equations under the assumption of $H^1$ solutions. For the Schrödinger equation we analyze the exponential-type scheme proposed by Ostermann and Schratz in 2018, whereas in the wave case we treat the corrected Lie splitting proposed by Li, Schratz, and Zivcovich in 2023. We show that the integrators converge with their full order of one and two, respectively. In this situation only fractional convergence rates were previously known. The crucial ingredients in the proofs are known space-time bounds for the solutions to the corresponding linear problems. More precisely, in the Schrödinger case we use the $L^4$ Strichartz inequality, and for the wave equation a null form estimate. To our knowledge, this is the first time that a null form estimate is exploited in numerical analysis. We apply the estimates for continuous time, thus avoiding potential losses resulting from discrete-time estimates.

Improved error estimates for low-regularity integrators using space-time bounds

TL;DR

This work establishes optimal convergence rates for low-regularity time integrators applied to the 1D periodic nonlinear Schrödinger and wave equations with initial data in . By deriving local-error representations and exploiting continuous-time space-time bounds—specifically the Strichartz estimate for Schrödinger and a novel null-form bound for the wave equation—the authors prove first-order convergence for the Ostermann–Schratz exponential-type integrator and second-order convergence for the corrected Lie splitting. These results surpass previously known fractional rates in this setting and rely on continuous-time estimates to avoid discretization-induced losses. The framework is flexible and may extend to higher dimensions or fully discrete schemes, highlighting the potential of space-time analysis in numerical treatment of dispersive and hyperbolic equations.

Abstract

We prove optimal convergence rates for certain low-regularity integrators applied to the one-dimensional periodic nonlinear Schrödinger and wave equations under the assumption of solutions. For the Schrödinger equation we analyze the exponential-type scheme proposed by Ostermann and Schratz in 2018, whereas in the wave case we treat the corrected Lie splitting proposed by Li, Schratz, and Zivcovich in 2023. We show that the integrators converge with their full order of one and two, respectively. In this situation only fractional convergence rates were previously known. The crucial ingredients in the proofs are known space-time bounds for the solutions to the corresponding linear problems. More precisely, in the Schrödinger case we use the Strichartz inequality, and for the wave equation a null form estimate. To our knowledge, this is the first time that a null form estimate is exploited in numerical analysis. We apply the estimates for continuous time, thus avoiding potential losses resulting from discrete-time estimates.

Paper Structure

This paper contains 5 sections, 11 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. The Schrödinger case
  3. The wave case
  4. Proof of the result for the nonlinear Schrödinger equation
  5. Proof of the result for the nonlinear wave equation

Key Result

Theorem 1.1

Let $r>1/2$ and $\gamma \in (0,1]$. Assume that the solution $u$ to NLS satisfies $u(t) \in H^{r+\gamma}(\mathbb{T})$ for all $t \in [0,T]$. Then there are a constant $C>0$ and a maximum step size $\tau_0>0$ such that the approximations $u_n$ obtained by LRI satisfy the error bound for all $\tau \in (0,\tau_0]$ and $n \in \mathbb{N}_0$ with $n\tau \le T$. The numbers $C$ and $\tau_0$ only depend

Theorems & Definitions (21)

  • Theorem 1.1: OS
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • proof : Proof of Theorem \ref{['ThmLRI']}
  • ...and 11 more