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**1 - 2**of**2**### Convergence rates for dispersive approximation schemes to nonlinear Schrödinger equations

- J. Math. Pures Appl
, 2012

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### NUMERICAL SCHEMES FOR THE NONLINEAR SCHRÖDINGER EQUATION

"... Abstract. We consider semidiscrete approximation schemes for the linear Schrödinger equation and analyze whether the classical dispersive properties of the continuous model hold for these approximations. For the conservative finite difference semi-discretization scheme of the linear Schrödinger eq ..."

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Abstract. We consider semidiscrete approximation schemes for the linear Schrödinger equation and analyze whether the classical dispersive properties of the continuous model hold for these approximations. For the conservative finite difference semi-discretization scheme of the linear Schrödinger equation, we show that, as the mesh-size tends to zero, the semidiscrete approximate solutions loose the dispersion property. We prove this property 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 three numerical remedies: Fourier filtering; numerical viscosity; two-grid preconditioner. For each of them we prove Strichartz-like estimates and the local space smoothing effect, uniformly on the mesh size. The methods we employ are based on classical estimates for oscillatory integrals. These estimates allow us to treat nonlinear problems with L2-initial data, without additional regularity hypotheses. We prove the convergence of the proposed methods for nonlinearities that cannot be handled by energy arguments and which, even in the continuous case, require Strichartz estimates. 1.