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Geometric Numerical Integration and Schrödinger Equations
Erwan Faou, ENS Cachan Bretagne, Bruz, France
A publication of the European Mathematical Society.
cover
Zurich Lectures in Advanced Mathematics
2012; 146 pp; softcover
Volume: 15
ISBN-10: 3-03719-100-7
ISBN-13: 978-3-03719-100-2
List Price: US$38
Member Price: US$30.40
Order Code: EMSZLEC/15
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The goal of geometric numerical integration is the simulation of evolution equations possessing geometric properties over long periods of time. Of particular importance are Hamiltonian partial differential equations typically arising in application fields such as quantum mechanics or wave propagation phenomena. They exhibit many important dynamical features such as energy preservation and conservation of adiabatic invariants over long periods of time. In this setting, a natural question is how and to which extent the reproduction of such long-time qualitative behavior can be ensured by numerical schemes.

Starting from numerical examples, these notes provide a detailed analysis of the Schrödinger equation in a simple setting (periodic boundary conditions, polynomial nonlinearities) approximated by symplectic splitting methods. Analysis of stability and instability phenomena induced by space and time discretization are given, and rigorous mathematical explanations are provided for them.

The book grew out of a graduate-level course and is of interest to researchers and students seeking an introduction to the subject matter.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

Readership

Graduate students and research mathematicians interested in geometric numerical integration, symplectic integrators, backward error analysis, and Schrödinger equations.

Table of Contents

  • Introduction
  • Finite-dimensional backward error analysis
  • Infinite-dimensional and semi-discrete Hamiltonian flow
  • Convergence results
  • Modified energy in the linear case
  • Modified energy in the semi-linear case
  • Introduction to long-time analysis
  • Bibliography
  • Index
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