On the Fourier Collocation Method
(2018) In Bachelor's Theses in Mathematical Sciences NUMK01 20181Mathematics (Faculty of Engineering)
- Abstract
- This BSc thesis focuses on trying to find approximate solutions to partial differential equations using the Fourier collocation method. This method uses Fourier basis functions to approximate the solution to a partial differential equation with periodic boundary conditions. Using Fourier basis functions, one does not need to use large matrices, which makes all computations relatively fast. Another benefit is that for smooth enough initial functions, the error converges very fast. We review some theory of Fourier basis functions, some theory of circulant matrices, and the theory underlying the implementation of the Fourier collocation method. We also describe how one can improve the error of the solution by making some extra calculations... (More)
- This BSc thesis focuses on trying to find approximate solutions to partial differential equations using the Fourier collocation method. This method uses Fourier basis functions to approximate the solution to a partial differential equation with periodic boundary conditions. Using Fourier basis functions, one does not need to use large matrices, which makes all computations relatively fast. Another benefit is that for smooth enough initial functions, the error converges very fast. We review some theory of Fourier basis functions, some theory of circulant matrices, and the theory underlying the implementation of the Fourier collocation method. We also describe how one can improve the error of the solution by making some extra calculations before applying the Fourier collocation method, and describe three methods of time stepping. In this BSc thesis, the advection-diffusion equation is used for testing the method, as it can be explicitly solved. We finally present some numerical results. These results confirm in large parts what the theory predicts. However, for some initial functions, the error does not converge as one would expect. (Less)
- Popular Abstract (Swedish)
- Partiella differentialekvationer är ekvationer som beskriver hur tillstånd förändrar sig. Dessa uppstår naturligt när man modellerar omvärlden, från hur molekyler sprider sig i celler till hur galaxer roterar. Att lösa dessa ekvationer är ofta näst intill omöjligt, och man behöver därför approximera lösningar med hjälp av datorer. Det här arbetet fokuserar på en speciell metod för att approximera lösningar, Fourier kollokationsmetoden. Denna metoden använder sig av periodiska funktioner (funktioner som upprepar sig med jämna mellanrum). Om man använder dessa funktioner blir alla beräkningar förhållandevis snabba, vilket är önskvärt. I detta arbete beskriver vi den matematiska teorin som underligger metoden. Vi beskriver sedan metoden och... (More)
- Partiella differentialekvationer är ekvationer som beskriver hur tillstånd förändrar sig. Dessa uppstår naturligt när man modellerar omvärlden, från hur molekyler sprider sig i celler till hur galaxer roterar. Att lösa dessa ekvationer är ofta näst intill omöjligt, och man behöver därför approximera lösningar med hjälp av datorer. Det här arbetet fokuserar på en speciell metod för att approximera lösningar, Fourier kollokationsmetoden. Denna metoden använder sig av periodiska funktioner (funktioner som upprepar sig med jämna mellanrum). Om man använder dessa funktioner blir alla beräkningar förhållandevis snabba, vilket är önskvärt. I detta arbete beskriver vi den matematiska teorin som underligger metoden. Vi beskriver sedan metoden och hur man kan förbättra approximationerna. Till sist presenterar vi resultatet från diverse numeriska experiment. Dessa bekräftar till stor del vad den underliggande teorin förutspår, men ett var förvånansvärt. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/8964033
- author
- Lindell, Henrik LU
- supervisor
- organization
- course
- NUMK01 20181
- year
- 2018
- type
- M2 - Bachelor Degree
- subject
- keywords
- Numerical Analysis, PDE, Partial Differential Equations, Numerical Linear Algebra, Mathematics, Fourier, Collocation, Numerical, Linear Algebra
- publication/series
- Bachelor's Theses in Mathematical Sciences
- report number
- LUNFNA-4024-2018
- ISSN
- 1654-6229
- other publication id
- 2018:K27
- language
- English
- id
- 8964033
- date added to LUP
- 2018-12-17 14:01:40
- date last changed
- 2018-12-17 14:01:40
@misc{8964033,
abstract = {{This BSc thesis focuses on trying to find approximate solutions to partial differential equations using the Fourier collocation method. This method uses Fourier basis functions to approximate the solution to a partial differential equation with periodic boundary conditions. Using Fourier basis functions, one does not need to use large matrices, which makes all computations relatively fast. Another benefit is that for smooth enough initial functions, the error converges very fast. We review some theory of Fourier basis functions, some theory of circulant matrices, and the theory underlying the implementation of the Fourier collocation method. We also describe how one can improve the error of the solution by making some extra calculations before applying the Fourier collocation method, and describe three methods of time stepping. In this BSc thesis, the advection-diffusion equation is used for testing the method, as it can be explicitly solved. We finally present some numerical results. These results confirm in large parts what the theory predicts. However, for some initial functions, the error does not converge as one would expect.}},
author = {{Lindell, Henrik}},
issn = {{1654-6229}},
language = {{eng}},
note = {{Student Paper}},
series = {{Bachelor's Theses in Mathematical Sciences}},
title = {{On the Fourier Collocation Method}},
year = {{2018}},
}