Generalised summation-by-parts operators and entropy stability of numerical methods for hyperbolic balance laws

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This thesis investigates and develops numerical methods for hyperbolic partial differential equations in continuum physics, presenting novel algorithms that are stable and robust. It extends the theory of conservative discretisations through summation-by-parts operators and symmetric numerical fluxes, applying these methods to nonlinear balance laws, including the shallow water and Euler equations. While formulating entropy stable schemes for the Euler equations with general summation-by-parts operators remains uncertain, classical operators can be utilized to construct such schemes. The research explores various numerical methods, developing new approaches that mimic continuous properties discretely. Additionally, the stability of fully discrete schemes using explicit Runge-Kutta methods is examined. A key concept throughout the investigations is the role of entropy in hyperbolic balance laws, which serves as a design principle for the numerical methods discussed. The thesis further extends these studies by investigating variational principles for entropy and their applicability in numerical schemes. This comprehensive work contributes significantly to the field, offering both theoretical insights and practical advancements in numerical analysis for continuum physics.