1Dispersive Properties of Numerical Schemes for Nonlinear Schrödinger Equations

In this article we report on recent work on building numerical approxi-mation schemes for nonlinear Schrödinger equations. We first consider finite-difference space semi-discretizations and show that the standard conservative scheme does not reproduce at the discrete level the disper-sion properties of the continuous Schrödinger equation. This is due to high frequency numerical spurious solutions. In order to damp out or filter these high-frequencies and to reflect the properties of the contin-uous problem we propose two remedies. First, adding a suitable extra numerical viscosity term at a convenient scale, and, second, a two-grid filter of the initial datum with meshes of ratio 1/4. We prove that these alternate schemes preserve the dispersion properties of the continuous model. We also present some applications to the numerical approxima-tion of nonlinear Schrödinger equations with initial data in L2. Despite the fact that classical energy methods fail, using these dispersion prop-erties, the numerical solutions of the semi-discrete nonlinear problems are proved to converge to the solution of the nonlinear Schrödinger equa-tion. We also discuss some open problems and some possible directions of future research. 1.