A Wigner-Measure Analysis of the Dufort--Frankel Scheme for the Schrödinger Equation

Abstract

Abstract. We apply Wigner transform techniques to the analysis of the Dufort-Frankel difference scheme for the Schrödinger equation and to the continuous analogue of the scheme in the case of a small (scaled) Planck constant (semiclassical regime). In this way we are able to obtain sharp conditions on the spatial-temporal grid which guarantee convergence for average values of observables as the Planck constant tends to zero. The theory developed in this paper is not based on local and global error estimates and does not depend on whether or not caustics develop. Numerical test examples are presented to help interpret the theory and to compare the Dufort-Frankel scheme to other difference schemes for the Schrödinger equation.

On Time-Splitting Spectral Approximations for the Schrödinger Equation in the Semiclassical Regime

“…Even for stable discretization schemes (or under mesh size restrictions which guarantee stability), the oscillations may very well pollute the solution in such a way that the quadratic macroscopic quantities and other physical observables come out completely wrong unless the spatial-temporal oscillations are fully resolved numerically, i.e., using many grid points per wavelength of O(ε). In [14,15], Markowich et al ultilized the Wigner measure, which was used in analyzing the semiclassical limit for the IVP (1.1) and (1.2), to study the finite difference approximation to the Schrödinger equation with small ε. Their results show that, for the best combination of the time and space discretizations, one needs the following constraint in order to guarantee good approximations to all (smooth) observables for ε small [14,15]:…”

Numerical Methods for the Nonlinear Schrödinger Equation with Nonzero Far-field Conditions

“…If a constant α is added to the potential V , then the discrete wave functions U ε,n+1 j obtained from TS-Cosine4 or TS-Cosine2 get multiplied by the phase factor e −iα(n+1)k/ε , which leaves the discrete quadratic observables unchanged. This property does not hold for finite difference schemes [25,26].…”

“…Introduction. The specific problem we study numerically in this paper is that of the nonlinear Schrödinger equation (NLS) with a small (scaled) Planck constant ε (0 < ε ≪ 1) given by [23,28,11,26,20]:…”

“…In [25,26], Markowich et al studied the finite difference approximation of the linear Schrödinger equation with small ε and zero far-field condition. Their results show that, for the best combination of the time and space discretizations, one needs the following constraints in order to guarantee good approximations to all (smooth) observables for ε small [25,26]: mesh size h = o(ε) and time step k = o(ε). The same or more severe meshing constraint is required by the finite difference approximation of the NLS in the semi-classical regime with zero far-field condition, i.e.…”

Gaussian beam methods for the Schrodinger equation in the semi-classical regime: Lagrangian and Eulerian formulations

“…A mesh size of O(ε) is required when using the time-splitting spectral method [1] to simulate (1.1)-(1.2) directly. The mesh size (and the time step as well) becomes even worse, since they need to be as small as o(ε), if finite difference methods are used [21,22]. The mesh and time step restrictions of these methods make the computation of (1.1)-(1.2) extremely expensive, especially in high dimensions.…”