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Stable difference methods for block-structured adaptive grids
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, Computational Science.
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, Computational Science.
2011 (English)Report (Other academic)
Place, publisher, year, edition, pages
2011.
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2011-022
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-159854OAI: oai:DiVA.org:uu-159854DiVA: diva2:447190
Projects
eSSENCE
Available from: 2011-10-11 Created: 2011-10-11 Last updated: 2013-11-29Bibliographically approved
In thesis
1. 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
2. Towards an adaptive solver for high-dimensional PDE problems on clusters of multicore processors
Open this publication in new window or tab >>Towards an adaptive solver for high-dimensional PDE problems on clusters of multicore processors
2012 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

Accurate numerical simulation of time-dependent phenomena in many spatial dimensions is a challenging computational task apparent in a vast range of application areas, for instance quantum dynamics, financial mathematics, systems biology and plasma physics. Particularly problematic is that the number of unknowns in the governing equations (the number of grid points) grows exponentially with the number of spatial dimensions introduced, often referred to as the curse of dimensionality. This limits the range of problems that we can solve, since the computational effort and requirements on memory storage directly depend on the number of unknowns for which to solve the equations.

In order to push the limit of tractable problems, we are developing an implementation framework, HAParaNDA, for high-dimensional PDE-problems. By using high-order accurate schemes and adaptive mesh refinement (AMR) in space, we aim at reducing the number of grid points used in the discretization, thereby enabling the solution of larger and higher-dimensional problems. Within the framework, we use structured grids for spatial discretization and a block-decomposition of the spatial domain for parallelization and load balancing. For integration in time, we use exponential integration, although the framework allows the flexibility of other integrators to be implemented as well. Exponential integrators using the Lanzcos or the Arnoldi algorithm has proven a succesful and efficient approach for large problems. Using a truncation of the Magnus expansion, we can attain high levels of accuracy in the solution.

As an example application, we have implemented a solver for the time-dependent Schrödinger equation using this framework. We provide scaling results for small and medium sized clusters of multicore nodes, and show that the solver fulfills the expected rate of convergence.

Place, publisher, year, edition, pages
Uppsala University, 2012
Series
Information technology licentiate theses: Licentiate theses from the Department of Information Technology, ISSN 1404-5117 ; 2012-003
National Category
Computer Science Computational Mathematics
Research subject
Scientific Computing
Identifiers
urn:nbn:se:uu:diva-169259 (URN)
Supervisors
Projects
eSSENCE
Available from: 2012-03-09 Created: 2012-02-25 Last updated: 2017-08-31Bibliographically approved

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Gustafsson, MagnusNissen, AnnaKormann, Katharina

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