Numerical approximation of the Schrödinger equation with concentrated potential

L. Banjai, M. López-Fernández

Research output: Contribution to journalArticle

Abstract

We present a family of algorithms for the numerical approximation of the Schrödinger equation with potential concentrated at a finite set of points. Our methods belong to the so-called fast and oblivious convolution quadrature algorithms. These algorithms are special implementations of Lubich's Convolution Quadrature which allow, for certain applications in particular parabolic problems, to significantly reduce the computational cost and memory requirements. Recently it has been noticed that their use can be extended to some hyperbolic problems. Here we propose a new family of such efficient algorithms tailored to the features of the Green's function for Schrödinger equations. In this way, we are able to keep the computational cost and the storage requirements significantly below existing approaches. These features allow us to perform reliable numerical simulations for longer times even in cases where the solution becomes highly oscillatory or seems to develop finite time blow-up. We illustrate our new algorithm with several numerical experiments.

Original languageEnglish
Article number109155
JournalJournal of Computational Physics
Volume405
Early online date9 Dec 2019
DOIs
Publication statusPublished - 15 Mar 2020

Keywords

  • Boundary integral equations
  • Contour integral methods
  • Convolution quadrature
  • Fast and oblivious algorithms
  • Schrödinger equation

ASJC Scopus subject areas

  • Numerical Analysis
  • Modelling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics

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