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High Order Finite Difference Methods with Artificial Boundary Treatment in Quantum Dynamics
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)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 [en]
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: urn:nbn:se:uu:diva-159856ISBN: 978-91-554-8180-3 (print)OAI: oai:DiVA.org:uu-159856DiVA: diva2:447201
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
List of papers
1. An optimized perfectly matched layer for the Schrödinger equation
Open this publication in new window or tab >>An optimized perfectly matched layer for the Schrödinger equation
2011 (English)In: Communications in Computational Physics, ISSN 1815-2406, E-ISSN 1991-7120, Vol. 9, 147-179 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-129182 (URN)10.4208/cicp.010909.010410a (DOI)000287557000007 ()
Projects
eSSENCE
Available from: 2010-08-05 Created: 2010-08-06 Last updated: 2017-12-12Bibliographically approved
2. Error control for simulations of a dissociative quantum system
Open this publication in new window or tab >>Error control for simulations of a dissociative quantum system
2010 (English)In: Numerical Mathematics and Advanced Applications: 2009, Berlin: Springer-Verlag , 2010, 523-531 p.Conference paper, Published paper (Refereed)
Place, publisher, year, edition, pages
Berlin: Springer-Verlag, 2010
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-132929 (URN)10.1007/978-3-642-11795-4_56 (DOI)978-3-642-11794-7 (ISBN)
Projects
eSSENCE
Available from: 2010-10-29 Created: 2010-10-29 Last updated: 2012-09-27Bibliographically approved
3. A perfectly matched layer applied to a reactive scattering problem
Open this publication in new window or tab >>A perfectly matched layer applied to a reactive scattering problem
2010 (English)In: Journal of Chemical Physics, ISSN 0021-9606, E-ISSN 1089-7690, Vol. 133, 054306:1-11 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-129174 (URN)10.1063/1.3458888 (DOI)000281215000013 ()
Projects
eSSENCE
Available from: 2010-08-04 Created: 2010-08-06 Last updated: 2017-12-12Bibliographically approved
4. High order stable finite difference methods for the Schrödinger equation
Open this publication in new window or tab >>High order stable finite difference methods for the Schrödinger equation
2011 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2011-014
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-154360 (URN)
Projects
eSSENCE
Available from: 2011-05-28 Created: 2011-05-31 Last updated: 2011-11-18Bibliographically approved
5. Stability at nonconforming grid interfaces for a high order discretization of the Schrödinger equation
Open this publication in new window or tab >>Stability at nonconforming grid interfaces for a high order discretization of the Schrödinger equation
2011 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2011-017
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-157120 (URN)
Projects
eSSENCE
Available from: 2011-08-16 Created: 2011-08-16 Last updated: 2011-11-18Bibliographically approved
6. Stable difference methods for block-structured adaptive grids
Open this publication in new window or tab >>Stable difference methods for block-structured adaptive grids
2011 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2011-022
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-159854 (URN)
Projects
eSSENCE
Available from: 2011-10-11 Created: 2011-10-11 Last updated: 2013-11-29Bibliographically approved

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