Energy and Entropy Stable High-Order Discontinuous Galerkin Methods on Cut Meshes

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dc.contributor.advisorChan, Jesseen_US
dc.creatorTaylor, Christinaen_US
dc.date.accessioned2024-08-30T16:36:18Zen_US
dc.date.available2024-08-30T16:36:18Zen_US
dc.date.created2024-08en_US
dc.date.issued2024-08-09en_US
dc.date.submittedAugust 2024en_US
dc.date.updated2024-08-30T16:36:18Zen_US
dc.description.abstractIn this work we present two contributions to cut mesh methods: a provably energy stable discontinuous Galerkin (DG) method with state redistribution and an entropy stable DG scheme for hyperbolic conservation laws. Cut meshes schemes use a simple unfitted background mesh of standard elements that is then cut by embedded boundaries to yield a hybrid mesh of cut and uncut elements. While these methods are a convenient means of mesh generation for complex geometries, the arbitrary size and shape of cut elements pose two significant challenges: the small cell problem--where the arbitrary size/shape of cut elements can severely restrict the CFL condition--and the construction of stable and accurate schemes on cut elements. We address the small cell problem using state redistribution and prove it can be combined with an energy stable scheme without damaging energy stability. For entropy stability, we construct hybridized summation-by-parts operators to reduce guaranteeing entropy stability to constructing exact, positive weight quadrature rules on cut elements. We construct such quadrature on cut elements using subtriangulations and Carathéodory pruning. We numerically verify the high-order accuracy and energy/entropy conservation and stability of our schemes using the acoustic wave, shallow water, and compressible Euler equations.en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.citationTaylor, Christina. Energy and Entropy Stable High-Order Discontinuous Galerkin Methods on Cut Meshes. (2024). PhD diss., Rice University. https://hdl.handle.net/1911/117797en_US
dc.identifier.urihttps://hdl.handle.net/1911/117797en_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.subjectEntropy Stableen_US
dc.subjectEnergy Stableen_US
dc.subjectCut Meshesen_US
dc.subjectState Redistributionen_US
dc.subjectHybridized Summation by Partsen_US
dc.subjectSummation by Partsen_US
dc.subjectSkew-symmetric formulationen_US
dc.titleEnergy and Entropy Stable High-Order Discontinuous Galerkin Methods on Cut Meshesen_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.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophyen_US
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