Low regularity error estimates for the time integration of 2D NLS

2023/01/25 by Lun Ji, Ji, Lun, Alexander Ostermann +5 · 1 citation
Engineering · Mathematics · #35Q55 #65M12 #65M15 #Advanced Mathematical Physics Problems #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2301.10639

openalex publication_date 2023/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schrödinger equation on the two-dimensional torus \mathbbT². The scheme is analyzed in a framework of discrete Bourgain spaces, which allows us to consider initial data with low regularity; more precisely initial data in H^s(\mathbbT²) with s>0. In this way, the usual stability restriction to smooth Sobolev spaces with index s>1 is overcome. Rates of convergence of order τ^s/2 in L²(\mathbbT²) at this regularity level are proved. Numerical examples illustrate that these convergence results are sharp.

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• Computing rough solutions of the KdV equation below \bf L²

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