  The Active Flux scheme is a Finite Volume scheme with additional degrees of
freedom. It makes use of a continuous reconstruction and does not require a
Riemann solver. An evolution operator is used for the additional degrees of
freedom on the cell boundaries. This paper presents progress towards the
computation of one-dimensional, viscous, compressible flows using Active Flux
scheme. An evolution operator for both linear and nonlinear hyperbolic
conservation systems is presented and then a novel extension is made to include
source terms. Applications are made on the Euler equations and a hyperbolic
formulation of the diffusion equation. Lastly, for the compressible
Navier-Stokes equations, a hyperbolic formulation is presented together with a
novel operator splitting approach. These allow for the Active Flux evolution
operators to be applied to the numerical computation of viscous, compressible
flows.
