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

Maximilian Ruff, Roland Schnaubelt

TL;DR

The paper addresses rigorous time-integration error analysis for the semilinear wave equation on $\mathbb{R}^3$ with finite-energy data, covering energy-subcritical to energy-critical nonlinearities $\alpha\in[3,5]$. It introduces frequency-filtered Lie splitting and new time-discrete Strichartz estimates, including endpoint results with logarithmic losses, to obtain provable convergence rates in weak norms. For subcritical powers ($\alpha<5$), first-order convergence in $L^2\times\dot H^{-1}$ is shown; at the energy-critical power ($\alpha=5$) a forward-error bound is established, reflecting the dispersive-dominated regime. Additionally, a corrected Lie splitting for the cubic case ($\alpha=3$) with $K=\tau^{-3/2}$ achieves $\tfrac{3}{2}$-order convergence, highlighting the benefit of the correction term and endpoint estimates. Overall, the work provides the first error analysis for scaling-critical dispersive problems with finite-energy data and introduces a robust framework combining discrete Strichartz theory with frequency filtering for NLW time-stepping.

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.

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

TL;DR

The paper addresses rigorous time-integration error analysis for the semilinear wave equation on with finite-energy data, covering energy-subcritical to energy-critical nonlinearities . It introduces frequency-filtered Lie splitting and new time-discrete Strichartz estimates, including endpoint results with logarithmic losses, to obtain provable convergence rates in weak norms. For subcritical powers (), first-order convergence in is shown; at the energy-critical power () a forward-error bound is established, reflecting the dispersive-dominated regime. Additionally, a corrected Lie splitting for the cubic case () with achieves -order convergence, highlighting the benefit of the correction term and endpoint estimates. Overall, the work provides the first error analysis for scaling-critical dispersive problems with finite-energy data and introduces a robust framework combining discrete Strichartz theory with frequency filtering for NLW time-stepping.

Abstract

We study time integration schemes for -solutions to the energy-(sub)critical semilinear wave equation on . We show first-order convergence in for the Lie splitting and convergence order 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.
Paper Structure (8 sections, 30 theorems, 213 equations)

This paper contains 8 sections, 30 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Strichartz estimates
  3. Time-discrete estimates
  4. Endpoint estimates with logarithmic loss
  5. Nonlinear Wave equation
  6. Lie splitting
  7. The critical case
  8. Corrected Lie Splitting

Key Result

Theorem 1.1

Let $\alpha<5$ and $U = (u,\partial_tu) \in C([0,T],\dot H^1(\mathbb{R}^3)\times L^2(\mathbb{R}^3))$ solve the semilinear wave equation NLW2. Then there are a constant $C>0$ and a maximum step size $\tau_0>0$ such that the iterates $U_n$ of the filtered Lie splitting scheme LieFilt0 or LieFilt satis for all $\tau \in (0,\tau_0]$ and $n \in \mathbb{N}_0$ with $n\tau \le T$. The numbers $C$ and $\ta

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • ...and 52 more