Chan, Jesse2022-09-232023-02-012022-082022-05-09August 202Wu, Xinhui. "High order entropy stable discontinuous Galerkin methods for the shallow water equations: networks, quasi-1D flows, and positivity preservation." (2022) Diss., Rice University. <a href="https://hdl.handle.net/1911/113214">https://hdl.handle.net/1911/113214</a>.https://hdl.handle.net/1911/113214High order schemes are known to be unstable in the presence of shock discontinuities or under-resolved solution features for nonlinear conservation laws. They have traditionally required additional filtering, limiting, or artificial viscosity to avoid solution blow up. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi-discrete entropy inequality independently of discretization parameters. This thesis focuses on high order entropy stable discontinuous Galerkin schemes for the shallow water equations, and extends them in two primary directions. First, we extend high order entropy stable discontinuous Galerkin method for nonlinear conservation laws to both multi-dimensional domains and networks constructed from 1D domains. These methods utilize treatments of multi-dimensional interfaces and network junctions which retain entropy stability when coupling together entropy stable discretizations. Numerical experiments verify the stability of the proposed schemes, and comparisons with fully 2D implementations demonstrate the accuracy of each type of junction treatment. Next, we provide both continuous and semi-discrete entropy analyses for the quasi-1D shallow water equations and the quasi-1D compressible Euler equations, which model one-dimensional flows through channels or nozzles with varying width. Each quasi-1D formulation includes additional terms to account for varying channel or nozzle widths. These extra terms correspond to non-conservative terms in the original equations, and introduce asymmetry into the numerical fluxes. We design new entropy conservative fluxes for both sets of equations and supply proofs of entropy conservation on periodic quasi-1D domains. Furthermore, we show that the new entropy conservative fluxes for the shallow water equations are well-balanced for continuous bathymetry profiles. Finally, we present a high order entropy stable discontinuous Galerkin method for nonlinear shallow water equations on 2D triangular meshes which preserves the positivity of the water height for constant bathymetry. The scheme combines a low order invariant domain preserving method with a high order entropy stable method using convex limiting. This method is entropy stable and well-balanced for meshes which are fitted to continuous bathymetry profiles. Finally, we apply the proposed method to a realistic large scale simulation of the 1959 Malpasset Dam break to verify the robustness of the scheme.application/pdfengCopyright 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.numerical PDEentropy stablediscontinuous Galerkinshallow water equationsThesis2022-09-23