  Active Flux is a third order accurate numerical method which evolves cell
averages and point values at cell interfaces independently. It naturally uses a
continuous reconstruction, but is stable when applied to hyperbolic problems.
In this work, the Active Flux method is extended for the first time to a
nonlinear hyperbolic system of balance laws, namely to the shallow water
equations with bottom topography. We demonstrate how to achieve an Active Flux
method that is well-balanced, positivity preserving, and allows for dry states
in one spatial dimension. Because of the continuous reconstruction all these
properties are achieved using new approaches. To maintain third order accuracy,
we also propose a novel high-order approximate evolution operator for the
update of the point values. A variety of test problems demonstrates the good
performance of the method even in presence of shocks.
