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Generalised Summation-by-Parts Operators and Entropy Stability of Numerical Methods for Hyperbolic Balance Laws

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Generalised Summation-by-Parts Operators and Entropy Stability of Numerical Methods for Hyperbolic Balance Laws (English shop)

Hendrik Ranocha (Author)

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ISBN-13 (Hard Copy) 9783736997356
ISBN-13 (eBook) 9783736987357
Language English
Page Number 304
Lamination of Cover matt
Edition 1.
Publication Place Göttingen
Place of Dissertation Braunschweig
Publication Date 2018-02-19
General Categorization Dissertation
Departments Mathematics
Keywords hyperbolic balance laws, hyperbolic conservation laws, summation-by-parts, entropy stability, applied mathematics, discontinuous Galerkin, CFD, Runge-Kutta methods, positivity preservation
URL to External Homepage https://ranocha.de/
Description

This thesis is dedicated to the investigation and development of numerical methods for hyperbolic partial differential equations arising in continuum physics and contains several new theoretical and practical insights which have resulted in novel numerical algorithms that are provably stable and robust, presented here for the first time as a whole. After extending the theory of conservative discretisations using summation-by-parts operators and symmetric numerical fluxes, the application of these methods to nonlinear balance laws such as the shallow water equations and the Euler equations is studied. While it is not clear whether entropy stable schemes can be formulated in this way for the Euler equations and general summation-by-parts operators, it is possible to construct such schemes using classical summation-by-parts operators. Following again the idea to mimic properties of the continuous level discretely, several numerical methods are investigated and new ones are developed. Moreover, stability of fully discrete schemes using explicit Runge-Kutta methods is investigate. Finally, an underlying concept of the previous investigations is studied in detail. Since the entropy plays a crucial role in the theory of hyperbolic balance laws, it has been used as a design principle of numerical methods as described before. Extending these studies, variational principles for the entropy are investigated with respect to their applicability in numerical schemes.