  The Active Flux scheme is a Finite Volume scheme with additional point values
distributed along the cell boundary. It is third order accurate and does not
require a Riemann solver: the continuous reconstruction serves as initial data
for the evolution of the points values. The intercell flux is then obtained
from the evolved values along the cell boundary by quadrature. This paper
focuses on the conceptual extension of Active Flux to include source terms, and
thus for simplicity assumes the homogeneous part of the equations to be linear.
To a large part, the treatment of the source terms is independent of the choice
of the homogeneous part of the system. Additionally, only systems are
considered which admit characteristics (instead of characteristic cones). This
is the case for scalar equations in any number of spatial dimensions and
systems in one spatial dimension. Here, we succeed to extend the Active Flux
method to include (possibly nonlinear) source terms while maintaining third
order accuracy of the method. This requires a novel (approximate) operator for
the evolution of point values and a modified update procedure of the cell
average. For linear acoustics with gravity, it is shown how to achieve a
well-balanced / stationarity preserving numerical method.
