Discontinuous Galerkin Methods for Parabolic Partial Differential Equations with Random Input Data
| dc.contributor.advisor | Riviere, Beatrice M. | en_US |
| dc.contributor.committeeMember | Heinkenschloss, Matthias | en_US |
| dc.contributor.committeeMember | Symes, William W. | en_US |
| dc.contributor.committeeMember | Vannucci, Marina | en_US |
| dc.creator | Liu, Kun | en_US |
| dc.date.accessioned | 2013-09-16T15:51:28Z | en_US |
| dc.date.accessioned | 2013-09-16T15:51:33Z | en_US |
| dc.date.available | 2013-09-16T15:51:28Z | en_US |
| dc.date.available | 2013-09-16T15:51:33Z | en_US |
| dc.date.created | 2013-05 | en_US |
| dc.date.issued | 2013-09-16 | en_US |
| dc.date.submitted | May 2013 | en_US |
| dc.date.updated | 2013-09-16T15:51:34Z | en_US |
| dc.description.abstract | This thesis discusses and develops one approach to solve parabolic partial differential equations with random input data. The stochastic problem is firstly transformed into a parametrized one by using finite dimensional noise assumption and the truncated Karhunen-Loeve expansion. The approach, Monte Carlo discontinuous Galerkin (MCDG) method, randomly generates $M$ realizations of uncertain coefficients and approximates the expected value of the solution by averaging M numerical solutions. This approach is applied to two numerical examples. The first example is a two-dimensional parabolic partial differential equation with random convection term and the second example is a benchmark problem coupling flow and transport equations. I first apply polynomial kernel principal component analysis of second order to generate M realizations of random permeability fields. They are used to obtain M realizations of random convection term computed from solving the flow equation. Using this approach, I solve the transport equation M times corresponding to M velocity realizations. The MCDG solution spreads toward the whole domain from the initial location and the contaminant does not leave the initial location completely as time elapses. The results show that MCDG solution is realistic, because it takes the uncertainty in velocity fields into consideration. Besides, in order to correct overshoot and undershoot solutions caused by the high level of oscillation in random velocity realizations, I solve the transport equation on meshes of finer resolution than of the permeability, and use a slope limiter as well as lower and upper bound constraints to address this difficulty. Finally, future work is proposed. | en_US |
| dc.format.mimetype | application/pdf | en_US |
| dc.identifier.citation | Liu, Kun. "Discontinuous Galerkin Methods for Parabolic Partial Differential Equations with Random Input Data." (2013) Diss., Rice University. <a href="https://hdl.handle.net/1911/71989">https://hdl.handle.net/1911/71989</a>. | en_US |
| dc.identifier.slug | 123456789/ETD-2013-05-431 | en_US |
| dc.identifier.uri | https://hdl.handle.net/1911/71989 | en_US |
| dc.language.iso | eng | en_US |
| dc.rights | Copyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder. | en_US |
| dc.subject | Parabolic PDEs | en_US |
| dc.subject | Monte Carlo Discontinuous Galerkin | en_US |
| dc.subject | Locally mass conservation | en_US |
| dc.subject | Random input data | en_US |
| dc.subject | Kernel PCA | en_US |
| dc.subject | Random permeability | en_US |
| dc.subject | Darcy's Law | en_US |
| dc.subject | Coupled flow and transport | en_US |
| dc.title | Discontinuous Galerkin Methods for Parabolic Partial Differential Equations with Random Input Data | 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 | Engineering | en_US |
| thesis.degree.grantor | Rice University | en_US |
| thesis.degree.level | Doctoral | en_US |
| thesis.degree.name | Doctor of Philosophy | en_US |