High order operator-splitting method for the numerical time propagation of the full Boltzmann equation in the interaction picture
This thesis focuses on finding a suitable high order splitting method for the numerical time propagation of the Boltzmann equation in the interaction method. Using Runge-Kutta family of numerical methods to create an adaptive step method in the interaction method to solve both collision and transpor...
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| Format: | Final Year Project (FYP) |
| Language: | English |
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Nanyang Technological University
2023
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| Online Access: | https://hdl.handle.net/10356/166536 |
| _version_ | 1811677527046881280 |
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| author | Zadzaan Bin Hassan |
| author2 | Marco Battiato |
| author_facet | Marco Battiato Zadzaan Bin Hassan |
| author_sort | Zadzaan Bin Hassan |
| collection | NTU |
| description | This thesis focuses on finding a suitable high order splitting method for the numerical time propagation of the Boltzmann equation in the interaction method. Using Runge-Kutta family of numerical methods to create an adaptive step method in the interaction method to solve both collision and transport term of the Boltzmann equation. The numerical method of Runge-Kutta 4 and Dormand-Prince 54 is first derived in their step-adaptive version. Followed by implementing the interaction picture into their step adaptive version. By introducing an Ordinary Differential Equation with similar structure to the Boltzmann, the two methods can be tested to observe on their performance. Next, by comparing the Order of Convergence of Runge-Kutta 4 and Dormand-Prince 54 method, it is found that Dormand Prince 54 method performs more efficient and accurately solving for numerical solution to a Boltzmann-like Ordinary Differential Equation. Lastly, an introduction of using a Dormand-Prince 54 method to calculate the product of a vector and exponential matrix rather than approximating the exponential matrix is shown to shorten computational time. |
| first_indexed | 2024-10-01T02:38:47Z |
| format | Final Year Project (FYP) |
| id | ntu-10356/166536 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T02:38:47Z |
| publishDate | 2023 |
| publisher | Nanyang Technological University |
| record_format | dspace |
| spelling | ntu-10356/1665362023-05-08T15:38:24Z High order operator-splitting method for the numerical time propagation of the full Boltzmann equation in the interaction picture Zadzaan Bin Hassan Marco Battiato School of Physical and Mathematical Sciences marco.battiato@ntu.edu.sg Science::Mathematics::Applied mathematics::Numerical analysis Science::Physics::Atomic physics::Statistical physics This thesis focuses on finding a suitable high order splitting method for the numerical time propagation of the Boltzmann equation in the interaction method. Using Runge-Kutta family of numerical methods to create an adaptive step method in the interaction method to solve both collision and transport term of the Boltzmann equation. The numerical method of Runge-Kutta 4 and Dormand-Prince 54 is first derived in their step-adaptive version. Followed by implementing the interaction picture into their step adaptive version. By introducing an Ordinary Differential Equation with similar structure to the Boltzmann, the two methods can be tested to observe on their performance. Next, by comparing the Order of Convergence of Runge-Kutta 4 and Dormand-Prince 54 method, it is found that Dormand Prince 54 method performs more efficient and accurately solving for numerical solution to a Boltzmann-like Ordinary Differential Equation. Lastly, an introduction of using a Dormand-Prince 54 method to calculate the product of a vector and exponential matrix rather than approximating the exponential matrix is shown to shorten computational time. Bachelor of Science in Physics 2023-05-04T06:40:30Z 2023-05-04T06:40:30Z 2023 Final Year Project (FYP) Zadzaan Bin Hassan (2023). High order operator-splitting method for the numerical time propagation of the full Boltzmann equation in the interaction picture. Final Year Project (FYP), Nanyang Technological University, Singapore. https://hdl.handle.net/10356/166536 https://hdl.handle.net/10356/166536 en application/pdf Nanyang Technological University |
| spellingShingle | Science::Mathematics::Applied mathematics::Numerical analysis Science::Physics::Atomic physics::Statistical physics Zadzaan Bin Hassan High order operator-splitting method for the numerical time propagation of the full Boltzmann equation in the interaction picture |
| title | High order operator-splitting method for the numerical time propagation of the full Boltzmann equation in the interaction picture |
| title_full | High order operator-splitting method for the numerical time propagation of the full Boltzmann equation in the interaction picture |
| title_fullStr | High order operator-splitting method for the numerical time propagation of the full Boltzmann equation in the interaction picture |
| title_full_unstemmed | High order operator-splitting method for the numerical time propagation of the full Boltzmann equation in the interaction picture |
| title_short | High order operator-splitting method for the numerical time propagation of the full Boltzmann equation in the interaction picture |
| title_sort | high order operator splitting method for the numerical time propagation of the full boltzmann equation in the interaction picture |
| topic | Science::Mathematics::Applied mathematics::Numerical analysis Science::Physics::Atomic physics::Statistical physics |
| url | https://hdl.handle.net/10356/166536 |
| work_keys_str_mv | AT zadzaanbinhassan highorderoperatorsplittingmethodforthenumericaltimepropagationofthefullboltzmannequationintheinteractionpicture |