Since the celebrated work of Lax and Wendroff in the early 60's, it is admitted that the right setting for the discretisation of conservation laws is first to define a consistent numerical flux, and then a conservative discretisation of the system. Following Hou and le Floch, it is also known if this is violated, then one approximate a system with a 'source' term in the form of a Borel measure. Hence there is no hope, apparently.
In this talk, I will revisit these ideas, showing that if one emphasis on conservation at the element level (and not at the cell interface level as in the classical setting), one can have much more flexibility. I will show how to exploit this to construct systematically discretisation that are entropy stable, or to construct methods that starts genuinely from a non conservative form of the equations, though leading to provable converging methods (to the right solutions). Examples in Lagrangian hydrodynamics and multi component problems will be given, as well as entropy stable schemes independently of the quadrature formula that are used to define the initial scheme. I will also draw some perspectives.
This is a joint work with P. Baccigalupi and S. Tokareva from IMath, University of Zurich.