High order numerical methods for hyperbolic equations: superconvergence, and applications to δ-singularities and cosmology

Yang, Yang

Full Metadata

Overview

Year:: 2013
Contributor:: Yang, Yang (creator); Shu, Chi-Wang (Director); Hesthaven, Jan (Reader); Guzman, Johnny (Reader); Brown University. Applied Mathematics (sponsor)
Genre:: theses
Subject:: Discontinuous Galerkin method; superconvergence; δ-singularities; WENO scheme; Cosmology
Extent:: 16, 169 p.
DOI: https://doi.org/10.7301/Z0C827ND

Files

Description

Abstract:: Part I introduces the discontinuous Galerkin (DG) method for solving hyperbolic equations. The introduction and the DG scheme will be given in the ﬁrst two chapters respectively. In Chapter 3, we apply energy analysis and dual argument to<br/> develop error estimates for the semi-discrete DG scheme and investigate the optimal superconvergence at downwind-biased Radau points as well as the superconvergence of the cell averages. The rest of this part focuses on the approximation of δ-functions. In Chapter 4, we consider negative-order norm error estimates on the whole computational domain and the region away from the singularities. We “extract” the<br/> hidden accuracy in the smooth region by post-processing the numerical solutions with suitable kernels. Some nonlinear examples, such as Rendez-vous algorithms and pressureless Euler equations, will be given in Chapter 5 to demonstrate the advantages of the DG scheme.<br/> In Part II, we apply weighted essentially non-oscillatory (WENO) schemes to the radiative transfer equation of Lyα photons in spherical halo. Chapter 6 introduces the basic knowledge, including the radiative transfer equation, the Eddington approximation, the WENO solver and other computational techniques. Chapter 7 investigates the effects of dust on Lyα photons emergent from an optically thick medium. We show that dust causes neither narrowing nor widening of the width of the double peaked proﬁle, and the time scales of the Lyα photon transfer in the optically thick halo are also shown to be basically independent of the dust scattering. Chapter 8 studies the angular distribution of Lyα photons transferring in or emerging from an optically thick medium. We test the precision of the Eddington approximation in this chapter.
Notes:: Thesis (Ph.D. -- Brown University (2013)

Content

Access Conditions

Rights: In Copyright
Restrictions on Use: Collection is open for research.

Citation

Yang, Yang, "High order numerical methods for hyperbolic equations: superconvergence, and applications to δ-singularities and cosmology" (2013). Applied Mathematics Theses and Dissertations. Brown Digital Repository. Brown University Library. https://doi.org/10.7301/Z0C827ND

Relations

Collection:

Applied Mathematics Theses and Dissertations

Theses and Dissertations for the Applied Mathematics department.
...