High-order WENO finite-difference methods for hyperbolic nonconservative systems of Partial Differential Equations

This work aims to extend the well-known high-order WENO finite-difference methods for systems of conservation laws to nonconservative hyperbolic systems. The main difficulty of these systems both from the theoretical and the numerical points of view comes from the fact that the definition of weak solution is not unique: according to the theory developed by Dal Maso, LeFloch, and Murat in 1995, it depends on the choice of a family of paths. A general strategy is proposed here in which WENO operators are not only used to reconstruct fluxes but also the nonconservative products of the system. Moreover, if a Roe linearization is available, the nonconservative products can be computed through matrix-vector operations instead of path-integrals. The methods are extended to problems with source terms and two different strategies are introduced to obtain well-balanced schemes. These numerical schemes will be then applied to the two-layer shallow water equations in one- and two- dimensions to obtain high-order methods that preserve water-at-rest steady states.