A filtered Lie splitting scheme for the 3D wave maps equation converges in discrete Bourgain spaces for all subcritical H^s initial data with s > 3/2.
Improved error estimates for low-regularity integrators using space-time bounds
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abstract
We prove optimal convergence rates for certain low-regularity integrators applied to the one-dimensional periodic nonlinear Schr\"odinger and wave equations under the assumption of $H^1$ solutions. For the Schr\"odinger 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 Schr\"odinger 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.
years
2026 2verdicts
UNVERDICTED 2representative citing papers
A hybrid solver-neural framework achieves global error O(τ^γ ln(1/τ)) for nonlinear dispersive equations by training a lightweight network on the residual defect inside the solver loop while preserving uniform stability.
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A splitting scheme for the wave maps equation at low regularity
A filtered Lie splitting scheme for the 3D wave maps equation converges in discrete Bourgain spaces for all subcritical H^s initial data with s > 3/2.
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Hybrid Iterative Neural Low-Regularity Integrator for Nonlinear Dispersive Equations
A hybrid solver-neural framework achieves global error O(τ^γ ln(1/τ)) for nonlinear dispersive equations by training a lightweight network on the residual defect inside the solver loop while preserving uniform stability.