pith. sign in

arxiv: 2311.03245 · v2 · submitted 2023-11-06 · 🧮 math.NA · cs.NA· math.AP

Error analysis of the Lie splitting for semilinear wave equations with finite-energy solutions

Maximilian Ruff , Roland Schnaubelt This is my paper

Pith reviewed 2026-05-24 05:50 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.AP
keywords Lie splittingsemilinear wave equationerror analysisStrichartz estimatesconvergence ratesfinite energy solutionsscaling critical
0
0 comments X

The pith

Lie splitting converges at first order in L² for Ḣ¹ solutions of the semilinear wave equation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves convergence rates for the Lie splitting time-stepping scheme applied to Ḣ¹ solutions of the energy-subcritical and critical semilinear wave equation in three dimensions. It establishes first-order accuracy in the L² norm for the standard Lie splitting and order 3/2 for a corrected version. The proofs rely on newly derived discrete-time Strichartz estimates that handle the scaling-critical setting. A reader would care because these are the first error bounds obtained for numerical methods on dispersive PDEs at the critical regularity where energy methods alone are insufficient.

Core claim

We show first-order convergence in L² for the Lie splitting and convergence order 3/2 for a corrected Lie splitting for Ḣ¹-solutions to the semilinear wave equation. This is based on discrete-time Strichartz estimates, including one with a logarithmic correction for the forbidden endpoint case. The schemes and estimates incorporate frequency cut-offs.

What carries the argument

Discrete-time Strichartz estimates with frequency cut-offs that control the error propagation in the splitting schemes for critical nonlinearities.

If this is right

  • The Lie splitting can be applied to approximate finite-energy solutions with guaranteed first-order accuracy in L².
  • A simple correction to the splitting improves the rate to 3/2.
  • Discrete Strichartz estimates provide a viable tool for analyzing numerical schemes in scaling-critical regimes.
  • The results extend error analysis to problems where standard techniques do not apply due to criticality.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Removing the frequency cut-offs without changing the rates would allow direct application to the original continuous problem.
  • Similar discrete Strichartz techniques could be tested on other time integrators or higher-dimensional cases.
  • The logarithmic correction in the endpoint estimate suggests potential refinements for other endpoint Strichartz applications in numerics.

Load-bearing premise

The frequency cut-offs in the numerical schemes and Strichartz estimates can be justified or removed for the target Ḣ¹ solutions without reducing the stated convergence orders.

What would settle it

A computation of the error for the unmodified Lie splitting without frequency cut-off on a smooth solution, showing whether the observed rate remains one or drops.

read the original abstract

We study time integration schemes for $\dot H^1$-solutions to the energy-(sub)critical semilinear wave equation on $\mathbb{R}^3$. We show first-order convergence in $L^2$ for the Lie splitting and convergence order $3/2$ for a corrected Lie splitting. To our knowledge this includes the first error analysis performed for scaling-critical dispersive problems. Our approach is based on discrete-time Strichartz estimates, including one (with a logarithmic correction) for the case of the forbidden endpoint. Our schemes and the Strichartz estimates contain frequency cut-offs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper analyzes time integration schemes for Ḣ¹ solutions of the energy-(sub)critical semilinear wave equation on ℝ³. It claims first-order L² convergence for the Lie splitting and order 3/2 for a corrected Lie splitting, based on discrete-time Strichartz estimates (including an endpoint case with logarithmic correction). Both the schemes and the estimates incorporate frequency cut-offs.

Significance. If the frequency cut-offs can be removed or rigorously justified without loss of the stated rates for the unmodified continuous problem, the result would constitute the first error analysis for scaling-critical dispersive problems and would be a notable contribution to numerical analysis of nonlinear wave equations with finite-energy data.

major comments (1)
  1. [Abstract] Abstract: The schemes and the supporting discrete Strichartz estimates are constructed with frequency cut-offs. The central claim of convergence orders for Ḣ¹ solutions of the unmodified problem requires a justification that these cut-offs can be removed (or sent to infinity) while preserving the rates; without such a step the stated orders apply only to a regularized problem whose solutions may not capture the full target regularity class.
minor comments (1)
  1. [Abstract] The abstract states that the work includes 'the first error analysis performed for scaling-critical dispersive problems'; a brief comparison paragraph with prior error analyses for subcritical or non-dispersive cases would help situate the novelty.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. Below we provide a point-by-point response to the major comment and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The schemes and the supporting discrete Strichartz estimates are constructed with frequency cut-offs. The central claim of convergence orders for Ḣ¹ solutions of the unmodified problem requires a justification that these cut-offs can be removed (or sent to infinity) while preserving the rates; without such a step the stated orders apply only to a regularized problem whose solutions may not capture the full target regularity class.

    Authors: We acknowledge that our analysis and schemes incorporate frequency cut-offs, as already stated in the abstract. The referee correctly identifies that without additional justification for passing to the limit as the cutoff tends to infinity, the convergence rates are established for the regularized problem. To address this, we will revise the abstract to more precisely describe the results as applying to the frequency-cutoff Lie splitting schemes for the semilinear wave equation. Furthermore, we will add a remark explaining the necessity of the cut-offs for the discrete Strichartz estimates and note that extending the result to the unmodified schemes without cut-offs remains an open question that may require new techniques. This revision ensures the claims accurately reflect what is proved in the paper. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on external Strichartz estimates

full rationale

The paper performs an error analysis for Lie splitting schemes on energy-critical semilinear wave equations, deriving first-order L2 convergence and 3/2-order for the corrected variant via discrete-time Strichartz estimates (including endpoint with log correction). These estimates are invoked as independent tools rather than constructed from the paper's own fitted quantities or self-referential definitions. Frequency cut-offs are explicitly present in both schemes and estimates, but this is a limitation on applicability to the unmodified problem, not a circular reduction where a prediction equals its input by construction. No self-citation load-bearing steps, uniqueness theorems imported from the authors, or ansatzes smuggled via prior work are indicated. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The analysis rests on the existence of discrete-time Strichartz estimates (including the logarithmic endpoint version) that are not derived here.

axioms (1)
  • domain assumption Discrete-time Strichartz estimates hold for the Lie splitting, including a version with logarithmic correction for the forbidden endpoint.
    The error analysis is explicitly based on these estimates as stated in the abstract.

pith-pipeline@v0.9.0 · 5624 in / 1196 out tokens · 26227 ms · 2026-05-24T05:50:54.528433+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

  1. [1]

    Bahouri, J

    H. Bahouri, J. Chemin, and R. Danchin. Fourier Analysis and Nonlinear Partial Differential Equations. Springer, Berlin Heidelberg, 2011

    work page 2011

  2. [2]

    H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differenti al Equations . Springer, New York, 2011

    work page 2011

  3. [3]

    On averaged ex ponential integrators for semilinear Klein–Gordon equations with solutions of low-r egularity

    S. Buchholz, B. Dörich, and M. Hochbruck. “On averaged ex ponential integrators for semilinear Klein–Gordon equations with solutions of low-r egularity”. In: SN Partial Differ. Equ. Appl. 2 (2021), pp. 2662–2963

    work page 2021

  4. [4]

    Scattering and Uniform in Time Error Estimates for Splitting Method in NLS

    R. Carles and C. Su. “Scattering and Uniform in Time Error Estimates for Splitting Method in NLS”. In: Found. Comput. Math. 24 (2024), pp. 683–722

    work page 2024

  5. [5]

    On the splitting method for the nonlin ear Schrödinger equa- tion with initial data in H 1

    W. Choi and Y. Koh. “On the splitting method for the nonlin ear Schrödinger equa- tion with initial data in H 1”. In: Discrete Contin. Dyn. Syst. 41 (2021), pp. 3837– 3867

    work page 2021

  6. [6]

    Dispersion of small amplitu de solutions of the gener- alized Korteweg–de Vries equation

    F. Christ and M. Weinstein. “Dispersion of small amplitu de solutions of the gener- alized Korteweg–de Vries equation”. In: J. Funct. Anal. 100 (1991), pp. 87–109

    work page 1991

  7. [7]

    Maximal Functions Associated to Filtrations

    M. Christ and A. Kiselev. “Maximal Functions Associated to Filtrations”. In: J. Funct. Anal. 179.2 (2001), pp. 409–425

    work page 2001

  8. [8]

    Error analysis of trigonometric integrat ors for semilinear Klein-Gordon equations

    L. Gauckler. “Error analysis of trigonometric integrat ors for semilinear Klein-Gordon equations”. In: SIAM J. Numer. Anal. 53 (2015), pp. 1082–1106

    work page 2015

  9. [9]

    A splitting method for the nonlinear Schrödin ger equation

    L. Ignat. “A splitting method for the nonlinear Schrödin ger equation”. In: J. Dif- ferential Equations 250 (2011), pp. 3022–3046

    work page 2011

  10. [10]

    Fully discrete schemes for the Schrödinger e quation: Dispersive proper- ties

    L. Ignat. “Fully discrete schemes for the Schrödinger e quation: Dispersive proper- ties”. In: Math. Models Methods Appl. Sci. 17.4 (2007), pp. 597–591

    work page 2007

  11. [11]

    Numerical dispersive schemes f or the nonlinear Schrödinger equation

    L. Ignat and E. Zuazua. “Numerical dispersive schemes f or the nonlinear Schrödinger equation”. In: SIAM J. Numer. Anal. 47.2 (2009), pp. 1366–1390

    work page 2009

  12. [12]

    Global Solutions to Maxwell Equations in a Ferromagnetic Medium

    J. Joly, G. Métivier, and J. Rauch. “Global Solutions to Maxwell Equations in a Ferromagnetic Medium”. In: Ann. Henri Poincaré 1 (2000), pp. 307–340

    work page 2000

  13. [13]

    A second-order low -regularity correction of Lie splitting for the semilinear Klein–Gordon equation

    B. Li, K. Schratz, and F. Zivcovich. “A second-order low -regularity correction of Lie splitting for the semilinear Klein–Gordon equation”. I n: ESAIM Math. Model. Numer. Anal. 57.2 (2023), pp. 899–919

    work page 2023

  14. [14]

    On splitting methods for Schrödinger-Pois son and cubic nonlinear Schrödinger equations

    C. Lubich. “On splitting methods for Schrödinger-Pois son and cubic nonlinear Schrödinger equations”. In: Math. Comp. 77 (2008), pp. 2141–2153

    work page 2008

  15. [15]

    Error estima tes at low regularity of splitting schemes for NLS

    A. Ostermann, F. Rousset, and K. Schratz. “Error estima tes at low regularity of splitting schemes for NLS”. In: Math. Comp. 91.333 (2021), pp. 169–182

    work page 2021

  16. [16]

    Error estima tes of a Fourier integrator for the cubic Schrödinger equation at low regularity

    A. Ostermann, F. Rousset, and K. Schratz. “Error estima tes of a Fourier integrator for the cubic Schrödinger equation at low regularity”. In: Found. Comput. Math. 21 (2021), pp. 725–765

    work page 2021

  17. [17]

    Fourier inte grator for periodic NLS: low regularity estimates via discrete Bourgain spaces

    A. Ostermann, F. Rousset, and K. Schratz. “Fourier inte grator for periodic NLS: low regularity estimates via discrete Bourgain spaces”. In : J. Eur. Math. Soc. 25.10 (2023), pp. 3913–3952

    work page 2023

  18. [18]

    On uniqueness for semilinear wave equati ons

    F. Planchon. “On uniqueness for semilinear wave equati ons”. In: Math. Z. 244 (2003), pp. 587–599

    work page 2003

  19. [19]

    A general framework of low re gularity integrators

    F. Rousset and K. Schratz. “A general framework of low re gularity integrators.” In: SIAM J. Numer. Anal. 59.3 (2021), pp. 1735–1768

    work page 2021

  20. [20]

    C. Sogge. Lectures on Non-Linear Wave Equations . International Press, Somerville, 2013

    work page 2013

  21. [21]

    Discrete analogues in Harmoni c Analysis II: Fractional Integration

    E. Stein and S. Wainger. “Discrete analogues in Harmoni c Analysis II: Fractional Integration”. In: J. d’Analyse Math. 80 (2000), pp. 335–355

    work page 2000

  22. [22]

    A counterexample to an endpoint bilinear Stric hartz inequality

    T. Tao. “A counterexample to an endpoint bilinear Stric hartz inequality”. In: Elec- tron. J. Differ. Equ. 151 (2006), pp. 1–6

    work page 2006

  23. [23]

    T. Tao. Nonlinear Dispersive Equations: Local and Global Analysis . American Math- ematical Society, Providence, RI, 2006

    work page 2006

  24. [24]

    Spacetime bounds for the energy-critical nonl inear wave equation in three spatial dimensions

    T. Tao. “Spacetime bounds for the energy-critical nonl inear wave equation in three spatial dimensions”. In: Dyn. Partial Differ. Equ. 3.2 (2006), pp. 93–110

    work page 2006

  25. [25]

    Y. Wu. A modified splitting method for the cubic nonlinear Schrödin ger equation . Preprint. 2022. doi: 10.48550/arXiv.2212.09301. REFERENCES 43 Karlsruhe Institute of Technology, Department of Mathemati cs, Engler- straße 2, 76131 Karlsruhe, Germany Email address : maximilian.ruff@kit.edu Email address : schnaubelt@kit.edu

    work page doi:10.48550/arxiv.2212.09301 2022