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Note: thesis: Thesis (Ph. D.)--Brown University, 2017
Genre (aat): theses
Abstract: The positivity-preserving property is a highly desirable property when designing high order numerical methods for hyperbolic conservation laws, since negative values sometimes cause ill-posedness of the problem and blow-ups of the algorithms. The general framework for constructing positivity-preserving schemes for solving hyperbolic conservation laws have been proposed in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), pp.~3091--3120) and (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), pp.~8918--8934). In this dissertation, we extend this framework to DG methods with implicit discretizations and to DG methods for solving the relativistic hydrodynamics (RHD). Due to the the Courant-Friedrichs-Levis (CFL) number constraint, explicit DG methods are impractical for problems involving unstructured and extremely varying meshes or long-time simulations. Instead, implicit DG schemes are often popular in practice, especially in the computational fluid dynamics (CFD) community. In the first part of this dissertation, we develop a high-order positivity-preserving DG method with the backward Euler time discretization for solving conservation laws, basing on a generalization of the Zhang-Shu positivity-preserving limiter. Both the analysis and numerical experiments indicate that a lower bound for the CFL number is required to obtain the positivity-preserving property for the numerical schemes. The proposed method not only preserves the positivity of the numerical approximation without compromising the designed high-order accuracy, but also helps accelerate the convergence towards the steady-state solution and add robustness to the nonlinear solver. For RHD systems, the density and the pressure are positive physical quantities and the velocity is bounded by the speed of light. The violation of these bounds will result in ill-posedness of the problem and blow-up of the code, especially in extreme relativistic cases. It is usually hard to maintain these physical bounds without sacrificing the numerical accuracy. In the second part, we develop a bound-preserving DG method to solve RHD systems by extending the bound-preserving limiter for the non-relativistic hydrodynamics. The proposed method has the following features. It can theoretically guarantee to preserve the physical bounds for the numerical approximation and maintain its designed high order accuracy. Moreover, it renders $L^1$-stability to the numerical scheme. The robustness of the scheme is tested on various extreme relativistic cases, including relativistic jets.
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Identifier: DOI: 10.7301/Z0TM78K9
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Access Condition: restriction on access: Collection is open for research.
Type of Resource (primo): dissertations