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We show how to combine in a natural way (i.e. without any test nor switch)the conservative and non conservative formulations of an hyperbolic system thathas a conservative form. This is inspired from two different class of schemes:the Residual Distribution one \cite{MR4090481}, and the Active Fluxformulations \cite{AF1, AF3, AF4,AF5,RoeAF}. The solution is globallycontinuous, and as in the active flux method, described by a combination ofpoint values and average values. Unlike the "classical" active flux methods,the meaning of the pointwise and cella averaged degrees of freedom isdifferent, and hence follow different form of PDEs: it is a conservativeversion of the cell average, and a possibly non conservative one for thepoints. This new class of scheme is proved to satisfy a Lax-Wendroff liketheorem. We also develop a method to perform non linear stability. Weillustrate the behaviour on several benchmarks, some quite challenging.

A combination of Residual Distribution and the Active Flux formulations  or a new class of schemes that can combine several writings of the same  hyperbolic problem: application to the 1D Euler equations