Dissipation and entropy production of high-order numerical methods for hyperbolic conservation laws

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This project is targeted at the investigation of stability and dissipation of high-order numerical methods for hyperbolic balance laws using the concept of nonlinear entropy stability. The entropy rate criterion of Dafermos will be used as a basis for the development and investigation of numerical methods and specially constructed numerical entropy fluxes will be studied regarding their significance in getting information about the properties of numerical solutions.Firstly, Godunov's flux for systems will be investigated. This numerical flux satisfies for scalar conservation laws a variational principle regarding the entropy dissipation. Afterwards, nonoscillatory recovery schemes based on variational entropy principles will be constructed and compared with their classical counterparts. Such variational ideas will be used for known entropy stable schemes and the resulting methods will be investigated regarding their stability properties.Moreover, information gained from specially constructed numerical entropy fluxes will be used to enhance the quality and stability of numerical methods, e.g. to apply and control additional dissipation via artificial viscosity or filtering. Finally, variational ideas will be applied to time discretisations.