Design of a theoretical framework to derive high-order numerical methods for hyperbolic systems of PDEs that include nonconservative products and/or source terms with good mathematical properties

Abstract: Many geophysical flow models have the form of nonlinear hyperbolic PDE systems that include source terms and nonconservative products. Besides the well-known theoretical and numerical difficulties related to systems of conservation laws, like the appearance of shock waves, new issues arise for these systems: definition of weak solutions, design of high-order shock capturing schemes, numerical methods that preserve the stationary solutions (well-balanced methods), etc. A theoretical framework has been developed to design numerical methods for these systems