Discontinuous Galerkin Methods for Parabolic Partial Differential Equations with Random Input Data

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dc.contributor.advisorRiviere, Beatrice M.
dc.contributor.committeeMemberHeinkenschloss, Matthias
dc.contributor.committeeMemberSymes, William W.
dc.contributor.committeeMemberVannucci, Marina
dc.creatorLiu, Kun
dc.date.accessioned2013-09-16T15:51:28Z
dc.date.accessioned2013-09-16T15:51:33Z
dc.date.available2013-09-16T15:51:28Z
dc.date.available2013-09-16T15:51:33Z
dc.date.created2013-05
dc.date.issued2013-09-16
dc.date.submittedMay 2013
dc.date.updated2013-09-16T15:51:34Z
dc.description.abstractThis 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.
dc.format.mimetypeapplication/pdf
dc.identifier.citationLiu, 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>.
dc.identifier.slug123456789/ETD-2013-05-431
dc.identifier.urihttps://hdl.handle.net/1911/71989
dc.language.isoeng
dc.rightsCopyright 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.
dc.subjectParabolic PDEs
dc.subjectMonte Carlo Discontinuous Galerkin
dc.subjectLocally mass conservation
dc.subjectRandom input data
dc.subjectKernel PCA
dc.subjectRandom permeability
dc.subjectDarcy's Law
dc.subjectCoupled flow and transport
dc.titleDiscontinuous Galerkin Methods for Parabolic Partial Differential Equations with Random Input Data
dc.typeThesis
dc.type.materialText
thesis.degree.departmentComputational and Applied Mathematics
thesis.degree.disciplineEngineering
thesis.degree.grantorRice University
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy
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