High frequency wave packets for the Schr\"odinger equation and its numerical approximations

TL;DR

This paper constructs Gaussian wave packets for the Schrödinger equation and its discretization, demonstrating the non-uniform dispersive properties of numerical solutions and how bigrid filtering can restore uniformity.

Contribution

It introduces a method to analyze high frequency wave packets in Schrödinger equations and their numerical approximations, highlighting the effects of filtering mechanisms on dispersive properties.

Findings

Numerical solutions lack uniform dispersive properties without filtering.

Bigrid algorithms can recover dispersive uniformity.

Wave packets split and propagate depending on filtering implementation.

Abstract

We build Gaussian wave packets for the linear Schr\"odinger equation and its finite difference space semi-discretization and illustrate the lack of uniform dispersive properties of the numerical solutions as established in Ignat, Zuazua, Numerical dispersive schemes for the nonlinear Schr\"odinger equation, SIAM. J. Numer. Anal., 47(2) (2009), 1366-1390. It is by now well known that bigrid algorithms provide filtering mechanisms allowing to recover the uniformity of the dispersive properties as the mesh size goes to zero. We analyze and illustrate numerically how these high frequency wave packets split and propagate under these bigrid filtering mechanisms, depending on how the fine grid/coarse grid filtering is implemented.