High-order accurate entropy stable numercial schemes for hyperbolic conservation laws

The role of entropy for the stability of hyperbolic conservation laws is well-understood, and to some extent, also for firstand second-order accurate numerical schemes. The stability of higher-order accurate schemes, however, is to a large degree an open problem. In this thesis we adopt the framework of entropy stability as a design principle. We combine entropy conservative fluxes with appropriate diffusion operators, and to obtain high order accuracy, we consider diffusion operators using the ENO reconstruction procedure. We show that the ENO procedure satisfies the so-called sign property, which ensures entropy stability of our (arbitrarily) high-order accurate finite difference scheme. Moreover, we pose and argue for a conjecture on the total variation of the ENO reconstruction, which would imply a weak total variation bound for our scheme. For hyperbolic systems, the reconstruction is performed using a novel, computationally efficient characteristic-wise decomposition. The scheme is easily generalized to multi-dimensional systems on Cartesian meshes. To study the convergence properties of our scheme for scalar equations, we consider the framework of compensated compactness. Under the assumption that our conjecture on the total variation bound of ENO holds, we prove convergence to the entropy solution. For multidimensional hyperbolic systems we are able to prove convergence to an entropy measurevalued solution. The robustness, accuracy and computational efficiency of the scheme are demonstrated in a series of numerical experiments.