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

01/25/2023

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by   Lun Ji, et al.

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A filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schrödinger equation on the two-dimensional torus 𝕋^2. 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(𝕋^2) 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 L^2(𝕋^2) at this regularity level are proved. Numerical examples illustrate that these convergence results are sharp.