Fast construction of the discrete Green operator for a second order ordinary differential equation
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917862803767296 |
|---|---|
| author | Blechta, Jan Průša, Vít Trnka, Ladislav Tůma, Karel |
| author_facet | Blechta, Jan Průša, Vít Trnka, Ladislav Tůma, Karel |
| contents | We consider linear second order differential equation y''= f with zero Dirichlet boundary conditions. At the continuous level this problem is solvable using the Green function, and this technique has a counterpart on the discrete level. The discrete solution is represented via an application of a matrix -- the Green matrix -- to the discretised right-hand side, and we propose an algorithm for fast construction of the Green matrix. In particular, we discretise the original problem using the spectral collocation method based on the Chebyshev--Gauss--Lobatto points, and using the discrete cosine transformation we show that the corresponding Green matrix is fast to construct even for large number of collocation points/high polynomial degree. Furthermore, we show that the action of the discrete solution operator (Green matrix) to the corresponding right-hand side can be implemented in a matrix-free fashion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06242 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fast construction of the discrete Green operator for a second order ordinary differential equation Blechta, Jan Průša, Vít Trnka, Ladislav Tůma, Karel Numerical Analysis 41A50, 34B05 We consider linear second order differential equation y''= f with zero Dirichlet boundary conditions. At the continuous level this problem is solvable using the Green function, and this technique has a counterpart on the discrete level. The discrete solution is represented via an application of a matrix -- the Green matrix -- to the discretised right-hand side, and we propose an algorithm for fast construction of the Green matrix. In particular, we discretise the original problem using the spectral collocation method based on the Chebyshev--Gauss--Lobatto points, and using the discrete cosine transformation we show that the corresponding Green matrix is fast to construct even for large number of collocation points/high polynomial degree. Furthermore, we show that the action of the discrete solution operator (Green matrix) to the corresponding right-hand side can be implemented in a matrix-free fashion. |
| title | Fast construction of the discrete Green operator for a second order ordinary differential equation |
| topic | Numerical Analysis 41A50, 34B05 |
| url | https://arxiv.org/abs/2412.06242 |