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An optimized perfectly matched layer for the Schrödinger equation
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
2011 (English)In: Communications in Computational Physics, ISSN 1815-2406, Vol. 9, 147-179 p.Article in journal (Refereed) Published
Place, publisher, year, edition, pages
2011. Vol. 9, 147-179 p.
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-129182DOI: 10.4208/cicp.010909.010410aISI: 000287557000007OAI: oai:DiVA.org:uu-129182DiVA: diva2:337596
Projects
eSSENCE
Available from: 2010-08-05 Created: 2010-08-06 Last updated: 2012-11-26Bibliographically approved
In thesis
1. Absorbing boundary techniques for the time-dependent Schrödinger equation
Open this publication in new window or tab >>Absorbing boundary techniques for the time-dependent Schrödinger equation
2010 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

Chemical dissociation processes are important in quantum dynamics. Such processes can be investigated theoretically and numerically through the time-dependent Schrödinger equation, which gives a quantum mechanical description of molecular dynamics.

This thesis discusses the numerical simulation of chemical reactions involving dissociation. In particular, an accurate boundary treatment in terms of artificial, absorbing boundaries of the computational domain is considered. The approach taken here is based on the perfectly matched layer technique in a finite difference framework. The errors introduced due to the perfectly matched layer can be divided into two categories, the modeling error from the continuous model and numerical reflections that arise for the discretized problem. We analyze the different types of errors using plane wave analysis, and parameters of the perfectly matched layer are optimized so that the modeling error and the numerical reflections are of the same order. The level of accuracy is determined by estimating the order of the spatial error in the interior domain. Numerical calculations show that this procedure enables efficient calculations within a given accuracy. We apply our perfectly matched layer to a three-state system describing a one-dimensional IBr molecule subjected to a laser field and to a two-dimensional model problem treating dissociative adsorbtion and associative desorption of an H2 molecule on a solid surface. Comparisons made to standard absorbing layers in chemical physics prove our approach to be efficient, especially when high accuracy is of importance.

Place, publisher, year, edition, pages
Uppsala University, 2010
Series
Information technology licentiate theses: Licentiate theses from the Department of Information Technology, ISSN 1404-5117 ; 2010-001
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-113087 (URN)
Supervisors
Projects
eSSENCE
Available from: 2010-02-11 Created: 2010-01-25 Last updated: 2017-08-31Bibliographically approved
2. High Order Finite Difference Methods with Artificial Boundary Treatment in Quantum Dynamics
Open this publication in new window or tab >>High Order Finite Difference Methods with Artificial Boundary Treatment in Quantum Dynamics
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The investigation of the dynamics of chemical reactions, both from the theoretical and experimental side, has drawn an increasing interest since Ahmed H. Zewail was awarded the 1999 Nobel Prize in Chemistry for his work on femtochemistry. On the experimental side, new techniques such as femtosecond lasers and attosecond lasers enable laser control of chemical reactions. Numerical simulations serve as a valuable complement to experimental techniques, not only for validation of experimental results, but also for simulation of processes that cannot be investigated through experiments. With increasing computer capacity, more and more physical phenomena fall within the range of what is possible to simulate. Also, the development of new, efficient numerical methods further increases the possibilities.

The focus of this thesis is twofold; numerical methods for chemical reactions including dissociative states and methods that can deal with coexistence of spatial regions with very different physical properties. Dissociative chemical reactions are reactions where molecules break up into smaller components. The dissociation can occur spontaneously, e.g. by radioactive decay, or be induced by adding energy to the system, e.g. in terms of a laser field. Quantities of interest can for instance be the reaction probabilities of possible chemical reactions. This thesis discusses a boundary treatment model based on the perfectly matched layer (PML) approach to accurately describe dynamics of chemical reactions including dissociative states. The limitations of the method are investigated and errors introduced by the PML are quantified.

The ability of a numerical method to adapt to different scales is important in the study of more complex chemical systems. One application of interest is long-range molecules, where the atoms are affected by chemical attractive forces that lead to fast movement in the region where they are close to each other and exhibits a relative motion of the atoms that is very slow in the long-range region. A numerical method that allows for spatial adaptivity is presented, based on the summation-by-parts-simultaneous approximation term (SBP-SAT) methodology. The accuracy and the robustness of the numerical method are investigated.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2011. 48 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 864
Keyword
Schrödinger equation, finite difference methods, perfectly matched layer, summation-by-parts operators, adaptive discretization, stability
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-159856 (URN)978-91-554-8180-3 (ISBN)
Public defence
2011-11-25, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, 10:15 (English)
Opponent
Supervisors
Projects
eSSENCE
Available from: 2011-11-03 Created: 2011-10-11 Last updated: 2011-11-10Bibliographically approved

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Nissen, AnnaKreiss, Gunilla

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