Shock capturing and high-order methods for hyperbolic conservation laws
Jan Glaubitz
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Jan Glaubitz, Shock capturing and high-order methods for hyperbolic conservation laws (2020), Logos Verlag, Berlin, ISBN: 9783832587208
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Beschreibung / Abstract
This thesis is concerned with the numerical treatment of hyperbolic conservation laws. These play an important role in describing many natural phenomena. Challenges in their theoretical as well as numerical study stem from the fact that spontaneous shock discontinuities can arise in their solutions, even in finite time and smooth initial states.
Moreover, the numerical treatment of hyperbolic conservations laws involves many different fields from mathematics, physics, and computer science. As a consequence, this thesis also provides contributions to several different fields of research -- which are still connected by numerical conservation laws, however. These contributions include, but are not limited to, the construction of stable high order quadrature rules for experimental data, the development of new stable numerical methods for conservation laws, and the investigation and design of shock capturing procedures as a means to stabilize high order numerical methods in the presence of (shock) discontinuities.
Moreover, the numerical treatment of hyperbolic conservations laws involves many different fields from mathematics, physics, and computer science. As a consequence, this thesis also provides contributions to several different fields of research -- which are still connected by numerical conservation laws, however. These contributions include, but are not limited to, the construction of stable high order quadrature rules for experimental data, the development of new stable numerical methods for conservation laws, and the investigation and design of shock capturing procedures as a means to stabilize high order numerical methods in the presence of (shock) discontinuities.
Beschreibung
Jan Glaubitz was born in Braunschweig, Germany, in 1990 and completed his mathematical studies (B.Sc., 2014, M.Sc., 2016, Dr. rer. nat., 2019) at TU Braunschweig. In 2016, he received awards from the German Mathematical Society (DMV) for his master's thesis as well as from the Society of Financial and Economic Mathematics of Braunschweig (VBFWM). In 2017, he was honored with the teaching award "LehrLEO" for the best tutorial at TU Braunschweig. Since 2020, he holds a position as a postdoctoral researcher at Dartmouth College, NH, USA.
Inhaltsverzeichnis
- BEGINN
- 1 Introduction
- 2 Hyperbolic conservation laws
- 2.1 General form and examples
- 2.2 Classical solutions
- 2.3 Breakdown of classical solutions
- 2.4 Weak solutions
- 2.5 The Riemann problem
- 2.6 The role of viscosity
- 2.7 Entropy solutions
- 2.8 Some results for scalar conservation laws
- 3 Numerical preliminaries
- 3.1 Approximation and interpolation
- 3.2 Orthogonal polynomials
- 3.3 Numerical differentiation
- 3.4 Numerical integration
- 3.5 Time integration
- 4 Stable high order quadrature rules for experimental data I: Nonnegative weight functions
- 4.1 Motivation
- 4.2 Least squares quadrature rules
- 4.3 Stability of least squares quadrature rules
- 4.4 Numerical tests
- 4.5 Concluding thoughts and outlook
- 5 Stable high order quadrature rules for experimental data II: General weight functions
- 5.1 Motivation
- 5.2 Stability concepts for general weight functions
- 5.3 Stability of least squares quadrature rules
- 5.4 Nonnegative least squares quadrature rules
- 5.5 Numerical results
- 5.6 Concluding thoughts and outlook
- 6 High order numerical methods for conservation laws
- 6.1 Discontinuous Galerkin spectral element methods
- 6.2 Flux reconstruction methods
- 7 Two novel high order methods
- 7.1 Stable discretizations of DG methods on equidistant and scattered points
- 7.2 Stable radial basis function methods
- 8 Artificial viscosity methods
- 8.1 The idea behind artificial viscosity
- 8.2 State of the art
- 8.3 Conservation and stability properties
- 8.4 New viscosity distributions
- 8.5 Modal filtering
- 8.6 Discretization of artificial viscosity terms using SBP operators
- 8.7 Concluding thoughts
- 9 `1 regularization and high order edge sensors for enhanced discontinuous Galerkin methods
- 9.1 Why `1 regularization?
- 9.2 Preliminaries
- 9.3 Application of `1 regularization to discontinuous Galerkin methods
- 9.4 Numerical results
- 9.5 Concluding thoughts and outlook
- 10 Shock capturing by Bernstein polynomials
- 10.1 Bernstein polynomials and the Bernstein operator
- 10.2 The Bernstein procedure
- 10.3 Entropy, total variation, and monotone (shock) profiles
- 10.4 Numerical results
- 10.5 Concluding thoughts and outlook
- 11 High order edge sensor steered artificial viscosity operators
- 11.1 Generalized artificial viscosity operators
- 11.2 Conservation and dissipation - continuous setting
- 11.3 High order FD methods based on SBP operators
- 11.4 Conservation and dissipation - discrete setting
- 11.5 High order edge sensor steered artificial viscsosity operators
- 11.6 Numerical results
- 11.7 Concluding thoughts and outlook
- 12 Summary and outlook
- Bibliography
- Index
- Glossary