Dispersive Properties of Numerical Schemes for Nonlinear Schrödinger Equations

In this article we report on recent work on building numerical approximation 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 dispersion 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 continuous 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 approximation of nonlinear Schrödinger equations with initial data in L. Despite the fact that classical energy methods fail, using these dispersion properties, the numerical solutions of the semi-discrete nonlinear problems are proved to converge to the solution of the nonlinear Schrödinger equation. We also discuss some open problems and some possible directions of future research. 1.