Reconstruction Techniques and Finite Volume Schemes for Hyperbolic Conservation Laws

Abstract: This thesis concerns the numerical approximation of the solutions to hyperbolic conservation laws. In particular the research work focuses on reconstruction techniques; the reconstruction being the key ingredient in modern finite volume schemes aiming to increase spatial order of accuracy. To better conform to the nature of the solutions to the hyperbolic problems, the reconstructing function is non-polynomial; in contrast to other reconstructions this allows us to have a continuous function representation, possibly having an extremum, within each spatial cell without limiting slopes. The flexible and simple to use reconstruction enables in a novel manner the derivation of schemes that efficiently combine the properties of accuracy, resolution and damping of spurious oscillations. Furthermore, applicability of the reconstruction is not restricted to Cartesian meshes as demonstrated by numerically solving the Euler equations of gas dynamics on triangular meshes in the finite volume context.