Convergence Results and a New Preconditioner for Spectral Collocation in Time
| dc.contributor.advisor | Heinkenschloss, Matthias | en_US |
| dc.creator | Steinman, John D. | en_US |
| dc.date.accessioned | 2025-01-17T17:15:10Z | en_US |
| dc.date.created | 2024-12 | en_US |
| dc.date.issued | 2024-12-06 | en_US |
| dc.date.submitted | December 2024 | en_US |
| dc.date.updated | 2025-01-17T17:15:10Z | en_US |
| dc.description.abstract | Spectral collocation methods provide a systematic construction for the approximate solution of ordinary differential equations (ODEs) of arbitrarily high order. These methods approximate the solution with a piecewise polynomial, which is determined by requiring the residual of the ODE to vanish at collocation points. This thesis presents three algebraically equivalent forms of the collocation method corresponding to different choices of polynomial bases. The convergence of global collocation for linear problems is analyzed from the viewpoint of projection methods, in which the projection operator represents interpolation by polynomials. This analysis is extended to nonlinear problems using the Kantorovich Theorem. Finally, a new preconditioner is presented that facilitates the efficient implementation of Chebyshev collocation methods. Numerical experiments demonstrate that the solution time of preconditioned spectral collocation behaves like O(K log K), where K is the number of collocation points, allowing for solves with over a million points. | en_US |
| dc.embargo.lift | 2025-06-01 | en_US |
| dc.embargo.terms | 2025-06-01 | en_US |
| dc.format.mimetype | application/pdf | en_US |
| dc.identifier.uri | https://hdl.handle.net/1911/118215 | en_US |
| dc.language.iso | en | en_US |
| dc.subject | spectral collocation | en_US |
| dc.subject | convergence | en_US |
| dc.subject | preconditioning | en_US |
| dc.title | Convergence Results and a New Preconditioner for Spectral Collocation in Time | en_US |
| dc.type | Thesis | en_US |
| dc.type.material | Text | en_US |
| thesis.degree.department | Computational and Applied Mathematics | en_US |
| thesis.degree.discipline | Computational & Applied Math, Computational & Applied Math | en_US |
| thesis.degree.grantor | Rice University | en_US |
| thesis.degree.level | Masters | en_US |
| thesis.degree.name | Master of Arts | en_US |