High order operator splitting for the Boltzmann equation within the interaction picture

Project aims to improve on a feature of the C++ library TORTOISE, the full time propagation of the Boltzmann equation comprising of the transport and scattering terms. This project focuses on the transport term and strives to reduce the cost and error by performing time propagation in the interactio...

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Main Author: Song, Jinhua
Other Authors: Marco Battiato
Format: Final Year Project
Language:English
Published: Nanyang Technological University 2024
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Online Access:https://hdl.handle.net/10356/175686
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Institution: Nanyang Technological University
Language: English
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spelling sg-ntu-dr.10356-1756862024-05-06T15:36:54Z High order operator splitting for the Boltzmann equation within the interaction picture Song, Jinhua Marco Battiato School of Physical and Mathematical Sciences marco.battiato@ntu.edu.sg Physics Boltzmann equation Discontinuous Galerkin Dormand-Prince 5(4) Project aims to improve on a feature of the C++ library TORTOISE, the full time propagation of the Boltzmann equation comprising of the transport and scattering terms. This project focuses on the transport term and strives to reduce the cost and error by performing time propagation in the interaction picture. Obtaining an analytical solution to the Boltzmann equation is extremely difficult, and numerical methods are often used to approximate a solution. First half of the project aims to use the discontinuous Garlerkin method to discretise the spatial and velocity spaces into individual elements as an approximation of the density of particles. This approximation allows for easier computation, but also introduces jump discontinuity at the edges. Luckily, we know that the Dirac delta function is defined when integrated over its domain. Hence, the integral of the Boltzmann transport equation is also defined, and can be solved with integration by parts. Results of the integration tells us that it can be interpreted as a sum of the volume flux within the element and the surface flux at the edges. Thus, this project first involves the building of the explicit volume and surface flux matrices, then they are propagated with the Dormand-Prince Runge-Kutta method implemented in the interaction picture. The main issue with performing the propagation using conventional Dormand-Prince is that the transport operator requires a significantly smaller step size than the scattering operator to be stable. Implementation in the interaction picture will resolve this problem and we will observe that this will help to reduce the overall cost. Bachelor's degree 2024-05-03T02:45:34Z 2024-05-03T02:45:34Z 2024 Final Year Project (FYP) Song, J. (2024). High order operator splitting for the Boltzmann equation within the interaction picture. Final Year Project (FYP), Nanyang Technological University, Singapore. https://hdl.handle.net/10356/175686 https://hdl.handle.net/10356/175686 en application/pdf Nanyang Technological University
institution Nanyang Technological University
building NTU Library
continent Asia
country Singapore
Singapore
content_provider NTU Library
collection DR-NTU
language English
topic Physics
Boltzmann equation
Discontinuous Galerkin
Dormand-Prince 5(4)
spellingShingle Physics
Boltzmann equation
Discontinuous Galerkin
Dormand-Prince 5(4)
Song, Jinhua
High order operator splitting for the Boltzmann equation within the interaction picture
description Project aims to improve on a feature of the C++ library TORTOISE, the full time propagation of the Boltzmann equation comprising of the transport and scattering terms. This project focuses on the transport term and strives to reduce the cost and error by performing time propagation in the interaction picture. Obtaining an analytical solution to the Boltzmann equation is extremely difficult, and numerical methods are often used to approximate a solution. First half of the project aims to use the discontinuous Garlerkin method to discretise the spatial and velocity spaces into individual elements as an approximation of the density of particles. This approximation allows for easier computation, but also introduces jump discontinuity at the edges. Luckily, we know that the Dirac delta function is defined when integrated over its domain. Hence, the integral of the Boltzmann transport equation is also defined, and can be solved with integration by parts. Results of the integration tells us that it can be interpreted as a sum of the volume flux within the element and the surface flux at the edges. Thus, this project first involves the building of the explicit volume and surface flux matrices, then they are propagated with the Dormand-Prince Runge-Kutta method implemented in the interaction picture. The main issue with performing the propagation using conventional Dormand-Prince is that the transport operator requires a significantly smaller step size than the scattering operator to be stable. Implementation in the interaction picture will resolve this problem and we will observe that this will help to reduce the overall cost.
author2 Marco Battiato
author_facet Marco Battiato
Song, Jinhua
format Final Year Project
author Song, Jinhua
author_sort Song, Jinhua
title High order operator splitting for the Boltzmann equation within the interaction picture
title_short High order operator splitting for the Boltzmann equation within the interaction picture
title_full High order operator splitting for the Boltzmann equation within the interaction picture
title_fullStr High order operator splitting for the Boltzmann equation within the interaction picture
title_full_unstemmed High order operator splitting for the Boltzmann equation within the interaction picture
title_sort high order operator splitting for the boltzmann equation within the interaction picture
publisher Nanyang Technological University
publishDate 2024
url https://hdl.handle.net/10356/175686
_version_ 1800916263543767040