We derive a semidiscrete in time, piecewise smooth in space, local weak formulation of multi-dimensional balance laws. This may serve as a basis of finite-volume and discontinuous Galerkin schemes. In particular, it highlights the different roles played by the conservative Riemann-flux and the non-conservative source term. We use this framework to derive a variant of the recently proposed “conservative form” of one-dimensional balance laws due to Chertock, Herty & Ozcan. Finally, we extend the approach to networks of gas pipelines and derive a well-balanced scheme.
