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" Efficient High Order Methods for Atomistic Calculations "
Mumtaz, Faisal
Alharbi, Fahhad H.
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Latin Dissertation
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English
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| Record Number
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1104461
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TLpq2234295250
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| Main Entry
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Alharbi, Fahhad H.
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Mumtaz, Faisal
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| Title & Author
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Efficient High Order Methods for Atomistic Calculations\ Mumtaz, FaisalAlharbi, Fahhad H.
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| College
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Hamad Bin Khalifa University (Qatar)
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| Date
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2019
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2019
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| Degree
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Ph.D.
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| Page No
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158
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| Abstract
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In the recent years, atomistic calculations have had a significant impact on materials science with a strong emphasis toward materials design. These calculations provide the basis for systematic screening and detailed analysis of materials properties which is critical to sustainable technological advancement. Beside the developments on the physical foundations of both the empirical and ab initio atomic-scale calculations, there has been a lot of development on the numerical methods to solve mathematical models in real space and time, and this research focuses on the development of efficient high-order numerical methods to solve the resulted differential equations from these mathematical models. The conventional numerical methods for the atomistic calculations are based on Slater Type Orbitals (STOs) and Gaussian Type Orbitals (GTOs), which are "applied mathematically" incomplete sets. Hence, there is a need to develop new basis sets, which are complete, orthogonal able to handle unbounded domains and cover a wide range of decay rates. With this in mind, we have developed a complete mapped basis set (MBS) based on Fourier Sine Series. We developed a generalized coordinate transformation method allowing the development of highly efficient basis sets for atomic scale calculations. The initial validation test to analyze the performance of the developed basis set is the approximation efficiency at different decay rates. In our calculations, four types of physically recurring decaying behaviors are considered, which are: oscillating/non-oscillating exponential/algebraic decays. The results and the analyses show that properly designed high-order mapped basis sets can be effective tools to handle challenging physical problems on unbounded domains. After the successful development of the mapped basis set, we have implemented it using the spectral method (SM) to solve some of the mathematical models involved in the atomistic calculations. These mathematical models include Poisson, Helmholtz and Schrödinger equation. A comparison with the existing methods is performed to illustrate the performance accuracy and efficiency of the proposed method. All the results show that the developed method can be a useful tool in solving the differential equations resulting from the mathematical models of the electronic structure problems.
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Applied Mathematics
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Computational chemistry
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Computational physics
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