I have been studying ex11.c from ts/tutorials to understand how to solve an hyperbolic system of equations using PETSCFV. I first worked on the Euler equations for inviscid fluids and based on what ex11.c presents, I was able to add the right PETSc instructions in an already existing in-house code with different gas models  to solve the problems in parallel (MPI) and with the AMR capabilities offered by P4EST.

Now my goal is to move to Navier-Stokes equations. Theoretically the system is not completely hyperbolic and can be seen as one with an hyperbolic part (identical to the Euler equations) and a parabolic part coming from the RHS diffusion terms.

I have been looking into the manual and also the sources of PETSc around the DM, DMPlex, DS and FV classes but I could not find anything that speaks to me as "adding a RHS to an hyperbolic system of equations" or "adding a source term to an hyperbolic system of equations". What's more, that source term depends on the derivatives of the context variables ... 

I wanted to know if anyone maybe had a suggestion regarding this issue ?

Thank you very much in advance,