Low Regularity Full Error Estimates for the Cubic Nonlinear Schrödinger Equation

.For the numerical solution of the cubic nonlinear Schrödinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting scheme in time is considered. This scheme is shown to converge even for initial data with very low regularity. In particular, for data in \(H^s( T^2)\) , where \(s> 0\) , convergence of order \( O(τs/2+N⁻ˢ)\) is proved in \(L^2\) . Here \(τ\) denotes the time step size and \(N\) the number of Fourier modes considered. The proof of this result is carried out in an abstract framework of discrete Bourgain spaces; the final convergence result, however, is given in \(L^2\) . The stated convergence behavior is illustrated by several numerical examples.Keywordsnonlinear Schrödinger equationlow regularityconvergence of full dicretizationLie splittingBourgain techniquesMSC codes65M1265M1535Q55