Since the celebrated theorem of Lax and Wendroff, we know a necessary condition that any numerical scheme for hyperbolic problem should satisfy: it should be written in flux form. A variant can also be formulated for the entropy. Even though some schemes, as for example those using continuous finite element, do not formally cast into this framework, it is a very convenient one. In this paper, we revisit this, introduce a different notion of local conservation which contains the previous one in one space dimension, and explore its consequences. This gives a more flexible framework that allows to get, systematically, entropy stable schemes, entropy dissipative ones, or accomodate more constraints. In particular, we can show that continuous finite element method can be rewritten in the finite volume framework, and all the quantities involved are explicitly computable. We end by presenting the only counter example we are aware of, i.e a scheme that seems not to be rewritten as a finite volume scheme.

翻译：自Lax和Wendroff的著名定理以来，我们认识到双曲问题数值格式需满足一个必要条件：其必须写成通量形式。针对熵亦可给出变体形式。尽管某些格式（例如使用连续有限元法的格式）并未严格纳入该框架，但该框架仍极具便利性。本文重新审视这一问题，引入一种不同于以往的一维空间局部守恒概念，并探讨其推论。这一更灵活的框架能够系统性地构建熵稳定格式、熵耗散格式，或适应更多约束条件。特别地，我们证明连续有限元法可重写为有限体积框架，且其中涉及的所有量均可显式计算。最后，我们给出目前已知的唯一反例——即看似无法重写为有限体积格式的一种格式。