implicit coding

Endeve

I have written an implicit code to solve the set of hyperbolic equatuions describing the solar wind. The parameters solved for are particle density, flow velocity, plasma temperature and heat flow in a so-called eight moment description. The equations are only solved in one spatial dimension. The aim is to study steady state solutions. I solve time dependent equations and let them evolve in time until a steady state is reached. My question is how important is the treatment of the non-linear advective term? Is the steady state solution dependent on the treatment of this term? I have coded a first order upwind Godunov flux and compared it with a second order Van-leer flux. The steady state solutions are equal, but shocks are steeper when a second order flux is used. But the point is not to study shocks, but to evolove transients out of the system and reach a steady state. Have anyone compared the different treatments of the advective term for implicit schemes?