High Order Numerical Methods for Hyperbolic Equations: Bound-preserving and Riemann Invariant Based System Solvers

Xu, Ziyao

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Year:: 2023
Contributor:: Xu, Ziyao (creator); Shu, Chi-Wang (Advisor); Ainsworth, Mark (Reader); Guzmán, Johnny (Reader); Brown University. Department of Applied Mathematics (sponsor)
Genre:: theses
Subject:: Conservation laws (Mathematics)--Numerical solutions
Extent:: 8, 260 p.

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Abstract:: This dissertation consists of three topics on bound-preserving discontinuous Galerkin (DG) methods for time-dependent and stationary hyperbolic equations, and efficient finite difference weighted essentially non-oscillatory (WENO) schemes for hyperbolic systems. In Chapter 1, we propose third order bound-preserving DG schemes for scalar conservation laws and the Euler equations based on the Lax-Wendroff time discretization. We first establish the maximum-principle-satisfying DG scheme for scalar conservation laws in one dimension. The scheme develops the idea from the direct discontinuous Galerkin (DDG) method for heat equations to discretize high order spatial derivatives resulting from the Lax-Wendroff procedure. When it extends to multi-dimensions, we avoid the appearance of mixed derivatives in the numerical schemes based on carefully designed expansions of high order derivatives. The positivity-preserving schemes for the Euler equations are constructed in a similar manner. In Chapter 2, we follow the work of Yuan et al. (2016) and Ling et al. (2018) to investigate the positivity-preserving DG methods for stationary hyperbolic equations. High order conservative positivity-preserving DG methods for variable coefficient and nonlinear stationary hyperbolic equations in one dimension, and constant coefficient stationary hyperbolic equations in two and three dimensions are constructed, via suitable quadratures. In Chapter 3, we continue the study in Chapter 2 and clarify a more appropriate definition of mass conservation, rather than preserving cell averages, for stationary hyperbolic equations. The genuinely conservative high-order positivity-preserving DG methods based on the new definition are constructed, which are able to preserve the positivity of more general types of equations with much simpler implementations and easier proofs for the Lax-Wendroff theorem. Novel conservative positivity-preserving limiters are designed to accommodate for the new definition of conservation. In Chapter 4, we investigate local characteristic decomposition free WENO schemes for a special class of hyperbolic systems endowed with a coordinate system of Riemann invariants. We apply the WENO procedure to the coordinate system of Riemann invariants instead of the local characteristic variables to save the expensive computational cost on local characteristic decomposition but meanwhile, maintain the essentially non-oscillatory performance. The efficiency and good performance of our method are demonstrated by extensive numerical tests, which indicate the coordinate system of Riemann invariants is a good alternative to local characteristic variables for the WENO procedure with higher efficiency.
Notes:: Thesis (Ph. D.)--Brown University, 2023

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Xu, Ziyao, "High Order Numerical Methods for Hyperbolic Equations: Bound-preserving and Riemann Invariant Based System Solvers" (2023). Applied Mathematics Theses and Dissertations. Brown Digital Repository. Brown University Library. https://repository.library.brown.edu/studio/item/bdr:s3kpe4fs/

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Applied Mathematics Theses and Dissertations

Theses and Dissertations for the Applied Mathematics department.
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