Lectures on Spectral Methods
These are a set of interactive notes where I review the basics of collocation spectral methods. I started writing these notes as a self-training, to have some simple compendium of available techniques in Colab notebooks. However, I decided to write them in the form of a tutorial, in case somebody else could find them useful. I intend to continuously update them with extra material as I experiment with new examples.
The contents of the notes are largely based on Lloyd N. Trefethen's amazing book
Spectral Methods in MATLAB
Lloyd Nicholas Trefethen
<br>Society for Industrial and Applied Mathematics, Philadelphia, 2000<br>
https://people.maths.ox.ac.uk/trefethen/spectral.html<br>
as well as some examples here and there from my own research and other papers in Theoretical Physics and General Relativity.
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<img src="https://github.com/raimonluna/LecturesOnSpectralMethods/blob/main/misc/hexagons.png" width="850"/>
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<em>Example of a solution of a nonlinear partial differential equation. See Lecture 5 for more information.</em>
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Contents:
1. Fourier Spectral Grid
<a href="https://colab.research.google.com/github/raimonluna/LecturesOnSpectralMethods/blob/main/Lecture1_FourierSpectralGrid.ipynb">
<img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/>
</a>
2. Chebyshev Spectral Grid
<a href="https://colab.research.google.com/github/raimonluna/LecturesOnSpectralMethods/blob/main/Lecture2_ChebyshevSpectralGrid.ipynb">
<img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/>
</a>
3. Linear ODEs and Eiganvalue Problems
<a href="https://colab.research.google.com/github/raimonluna/LecturesOnSpectralMethods/blob/main/Lecture3_LinearODEs_Eigenvalues.ipynb">
<img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/>
</a>
4. Two-Variable Functions
<a href="https://colab.research.google.com/github/raimonluna/LecturesOnSpectralMethods/blob/main/Lecture4_TwoVariableFunctions_PDEs.ipynb">
<img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/>
</a>
5. Nonlinear Differential Equations
<a href="https://colab.research.google.com/github/raimonluna/LecturesOnSpectralMethods/blob/main/Lecture5_NonlinearDifferentialEquations.ipynb">
<img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/>
</a>
6. Time Evolution of Hyperbolic and Parabolic Equations
<a href="https://colab.research.google.com/github/raimonluna/LecturesOnSpectralMethods/blob/main/Lecture6_TimeEvolution.ipynb">
<img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/>
</a>
7. Multi-Domain Grids and Quasinormal Modes
<a href="https://colab.research.google.com/github/raimonluna/LecturesOnSpectralMethods/blob/main/Lecture7_Multidomain_Quasinormal_Modes.ipynb">
<img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/>
</a>
