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Rhebergen, S. (2010) Discontinuous Galerkin finite element methods for (non)conservative partial differential equations. PhD thesis, University of Twente. ISBN 978-90-365-2964-8
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Official URL: http://dx.doi.org/10.3990/1.9789036529648
The first research topic in this thesis is the development of discontinuous Galerkin (DG) finite element methods for partial differential equations containing nonconservative products, which are present in many two-phase flow models. For this, we combine the theory of Dal Maso, LeFloch and Murat, in which a definition is given for nonconservative products even where the solution field is discontinuous. This theory also provides the mathematical foundation for a new DG finite element method. For this new DG method, we show standard (p+1)-order convergence results using p-th order basis-functions for test-cases of which we know the exact solution. We also show its ability to deal with more complex test cases. Finally, we apply the method to a depth-averaged two-phase flow model of which the numerical results are qualitatively validated against results obtained from a laboratory experiment.
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