  We propose an arbitrarily high-order accurate numerical method for
conservation laws that is based on a continuous approximation of the solution.
The degrees of freedom are point values at cell interfaces and moments of the
solution inside the cell. To lowest ($3^\text{rd}$) order this method reduces
to the Active Flux method. The update of the moments is achieved immediately by
integrating the conservation law over the cell, integrating by parts and
employing the continuity across cell interfaces. We propose two ways how the
point values can be updated in time: either by first deriving a semi-discrete
method that uses a finite-difference-type formula to approximate the spatial
derivative, and integrating this method e.g. with a Runge-Kutta scheme, or by
using a characteristics-based update, which is inspired by the original (fully
discrete) Active Flux method. We analyze stability and accuracy of the
resulting methods.
