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Stable difference methods for block-structured adaptive grids
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Avdelningen för teknisk databehandling. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Tillämpad beräkningsvetenskap.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Avdelningen för teknisk databehandling. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Numerisk analys.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Avdelningen för teknisk databehandling. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Tillämpad beräkningsvetenskap.
2011 (Engelska)Rapport (Övrigt vetenskapligt)
Abstract [en]

The time-dependent Schrödinger equation describes quantum dynamical phenomena. Solving it numerically, the small-scale interactions that are modeled require very fine spatial resolution. At the same time, the solutions are localized and confined to small regions in space. Using the required resolution over the entire high-dimensional domain often makes the model problems intractable due to the prohibitively large grids that result from such a discretization. In this paper, we present a block-structured adaptive mesh refinement scheme, aiming at efficient adaptive discretization of high-dimensional partial differential equations such as the time-dependent Schrödinger equation. Our framework allows for anisotropic grid refinement in order to avoid unnecessary refinement. For spatial discretization, we use standard finite difference stencils together with summation-by-parts operators and simultaneous-approximation-term interface treatment. We propagate in time using exponential integration with the Lanczos method. Our theoretical and numerical results show that our adaptive scheme is stable for long time integrations. We also show that the discretizations meet the expected convergence rates.

Ort, förlag, år, upplaga, sidor
2011.
Serie
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2011-022
Nationell ämneskategori
Beräkningsmatematik
Identifikatorer
URN: urn:nbn:se:uu:diva-159854OAI: oai:DiVA.org:uu-159854DiVA, id: diva2:447190
Projekt
eSSENCETillgänglig från: 2011-10-11 Skapad: 2011-10-11 Senast uppdaterad: 2024-05-30Bibliografiskt granskad
Ingår i avhandling
1. High Order Finite Difference Methods with Artificial Boundary Treatment in Quantum Dynamics
Öppna denna publikation i ny flik eller fönster >>High Order Finite Difference Methods with Artificial Boundary Treatment in Quantum Dynamics
2011 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Uppsala: Acta Universitatis Upsaliensis, 2011. s. 48
Serie
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 864
Nyckelord
Schrödinger equation, finite difference methods, perfectly matched layer, summation-by-parts operators, adaptive discretization, stability
Nationell ämneskategori
Beräkningsmatematik
Forskningsämne
Beräkningsvetenskap med inriktning mot numerisk analys
Identifikatorer
urn:nbn:se:uu:diva-159856 (URN)978-91-554-8180-3 (ISBN)
Disputation
2011-11-25, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, 10:15 (Engelska)
Opponent
Handledare
Projekt
eSSENCE
Tillgänglig från: 2011-11-03 Skapad: 2011-10-11 Senast uppdaterad: 2011-11-10Bibliografiskt granskad
2. Towards an adaptive solver for high-dimensional PDE problems on clusters of multicore processors
Öppna denna publikation i ny flik eller fönster >>Towards an adaptive solver for high-dimensional PDE problems on clusters of multicore processors
2012 (Engelska)Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Uppsala University, 2012
Serie
IT licentiate theses / Uppsala University, Department of Information Technology, ISSN 1404-5117 ; 2012-003
Nationell ämneskategori
Datavetenskap (datalogi) Beräkningsmatematik
Forskningsämne
Beräkningsvetenskap
Identifikatorer
urn:nbn:se:uu:diva-169259 (URN)
Handledare
Projekt
eSSENCEUPMARC
Tillgänglig från: 2012-03-09 Skapad: 2012-02-25 Senast uppdaterad: 2019-02-25Bibliografiskt granskad

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Gustafsson, MagnusNissen, AnnaKormann, Katharina

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