02538nam 22004335a 450 991015192800332120250513220847.03-03719-600-910.4171/100(CKB)3710000000953878(CH-001817-3)143-111229(PPN)178156035(EXLCZ)99371000000095387820111229j20120114 fy 0engurnn|mmmmamaatxtrdacontentcrdamediacrrdacarrierGeometric Numerical Integration and Schrödinger Equations /Erwan FaouZuerich, Switzerland European Mathematical Society Publishing House20121 online resource (146 pages)Zurich Lectures in Advanced Mathematics (ZLAM)The goal of geometric numerical integration is the simulation of evolution equations possessing geometric properties over long times. 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 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 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.Numerical analysisbicsscNumerical analysismscPartial differential equationsmscDynamical systems and ergodic theorymscNumerical analysisNumerical analysisPartial differential equationsDynamical systems and ergodic theory65-xx35-xx37-xxmscFaou Erwan1070685ch0018173BOOK9910151928003321Geometric Numerical Integration and Schrödinger Equations2564723UNINA