Numerical Dispersive Schemes for the Nonlinear Schrödinger Equation

We consider semidiscrete approximation schemes for the linear Schrodinger equation and analyze whether the classical dispersive properties of the continuous model hold for these approximations. For the conservative finite difference semidiscretization scheme we show that, as the mesh size tends to zero, the semidiscrete approximate solutions lose the dispersion property. This fact is proved by constructing solutions concentrated at the points of the spectrum where the second order derivatives of the symbol of the discrete Laplacian vanish. Therefore this phenomenon is due to the presence of numerical spurious high frequencies. To recover the dispersive properties of the solutions at the discrete level, we introduce two numerical remedies: Fourier filtering and a two-grid preconditioner. For each of them we prove Strichartz-like estimates and a local space smoothing effect, uniform in the mesh size. The methods we employ are based on classical estimates for oscillatory integrals. These estimates allow us to treat nonlinear problems with $L^2$-initial data, without additional regularity hypotheses. We prove the convergence of the two-grid method for nonlinearities that cannot be handled by energy arguments and which, even in the continuous case, require Strichartz estimates.