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This article is devoted to the construction of new numerical methods for the semiclassical Schrodinger equation. A phase-amplitude reformulation of the equation is described where the Planck constant epsilon is not a singular parameter. This allows to build splitting schemes whose accuracy is spectral in space, of up to fourth order in time, and independent of epsilon before the caustics. The second-order method additionally preserves the L^2-norm of the solution just as the exact flow does. In this first part of the paper, we introduce the basic splitting scheme in the nonlinear case, reveal our strategy for constructing higher-order methods, and illustrate their properties with simulations. In the second part, we shall prove a uniform convergence result for the first-order splitting scheme applied to the linear Schrodinger equation with a potential.
This article is devoted to the construction of numerical methods which remain insensitive to the smallness of the semiclassical parameter for the linear Schr{o}dinger equation in the semiclassical limit. We specifically analyse the convergence behavi
In this paper, we propose a numerical method to approximate the solution of the time-dependent Schrodinger equation with periodic boundary condition in a high-dimensional setting. We discretize space by using the Fourier pseudo-spectral method on ran
We consider the propagation of wave packets for the nonlinear Schrodinger equation, in the semi-classical limit. We establish the existence of a critical size for the initial data, in terms of the Planck constant: if the initial data are too small, t
We approximate the solution for the time dependent Schrodinger equation (TDSE) in two steps. We first use a pseudo-spectral collocation method that uses samples of functions on rank-1 or rank-r lattice points with unitary Fourier transforms. We then
We introduce a textit{non-modal} analysis technique that characterizes the diffusion properties of spectral element methods for linear convection-diffusion systems. While strictly speaking only valid for linear problems, the analysis is devised so th