Explicit Discontinuous Galerkin Methods for Linear Hyperbolic Problems

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dc.contributor.advisorWarburton, Timothyen_US
dc.contributor.committeeMemberSymes, William W.en_US
dc.contributor.committeeMemberEmbree, Marken_US
dc.creatorAtcheson, Thomasen_US
dc.date.accessioned2013-11-14T16:40:50Zen_US
dc.date.accessioned2013-11-14T16:40:54Zen_US
dc.date.available2013-11-14T16:40:50Zen_US
dc.date.available2013-11-14T16:40:54Zen_US
dc.date.created2013-12en_US
dc.date.issued2013-11-14en_US
dc.date.submittedDecember 2013en_US
dc.date.updated2013-11-14T16:40:57Zen_US
dc.description.abstractDiscontinuous Galerkin methods have many features which make them a natural candidate for the solution of hyperbolic problems. One feature is flexibility with the order of approximation; a user with knowledge of the solution's regularity can increase the spatial order of approximation by increasing the polynomial order of the discontinuous Galerkin method. A marked increase in time-stepping difficulty, known as stiffness, often accompanies this increase in spatial order however. This thesis analyzes two techniques for reducing the impact of this stiffness on total time of simulation. The first, operator modification, directly modifies the high order method in a way that retains the same formal order of accuracy, but reduces the stiffness. The second, optimal Runge-Kutta methods, adds additional stages to Runge-Kutta methods and modifies them to customize their stability region to the problem. Three operator modification methods are analyzed analytically and numerically, the mapping technique of Kosloff/Tal-Ezer the covolume filtering technique of Warburton/Hagstrom , and the flux filtering technique of Chalmers, et al. . The covolume filtering and flux filtering techniques outperform mapping in that they negligibly impact accuracy but yield a reasonable improvement in efficiency. For optimal Runge-Kutta methods this thesis considers five top performing methods from the literature on hyperbolic problems and applies them to an unmodified method, a flux filtered method, and a covolume filtered method. Gains of up to 80\% are seen for covolume filtered solutions applied with optimal Runge-Kutta methods, showing the potential for efficient high order solutions of unsteady systems.en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.citationAtcheson, Thomas. "Explicit Discontinuous Galerkin Methods for Linear Hyperbolic Problems." (2013) Master’s Thesis, Rice University. <a href="https://hdl.handle.net/1911/75120">https://hdl.handle.net/1911/75120</a>.en_US
dc.identifier.slug123456789/ETD-2013-12-584en_US
dc.identifier.urihttps://hdl.handle.net/1911/75120en_US
dc.language.isoengen_US
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.en_US
dc.subjectDiscontinuous Galerkinen_US
dc.subjectTime-steppingen_US
dc.subjectMathematicsen_US
dc.titleExplicit Discontinuous Galerkin Methods for Linear Hyperbolic Problemsen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentComputational and Applied Mathematicsen_US
thesis.degree.disciplineEngineeringen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelMastersen_US
thesis.degree.nameMaster of Artsen_US
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