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A Time-Space Adaptive Method for the Schrödinger Equation

Part of: Quantum theory Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  22 June 2016

Katharina Kormann
Katharina Kormann*
Affiliation:
Technische Universität München, Zentrum Mathematik, Boltzmannstr. 3, 85747 Garching, Germany
*
*Corresponding author. Email address:katharina.kormann@tum.de (K. Kormann)
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Abstract

In this paper, we present a discretization of the time-dependent Schrödinger equation based on a Magnus-Lanczos time integrator and high-order Gauss-Lobatto finite elements in space. A truncated Galerkin orthogonality is used to obtain duality-based a posteriori error estimates that address the temporal and the spatial error separately. Based on this theory, a space-time adaptive solver for the Schrödinger equation is devised. An efficient matrix-free implementation of the differential operator, suited for spectral elements, is used to enable computations for realistic configurations. We demonstrate the performance of the algorithm for the example of matter-field interaction.

Keywords

a posteriori error estimateadaptive mesh refinementfinite/spectral elementstime-dependent Schrödinger equation

MSC classification

Secondary: 65M50: Mesh generation and refinement 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65Z05: Applications to physics 81-08: Computational methods
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 1 , July 2016 , pp. 60 - 85
Copyright
Copyright © Global-Science Press 2016 

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