Linearization and continuation for upwind FVM schemes
 

Hello all,

The question in short: Is it common to get a discontinuous Jacobian at steady state solutions when using upwind schemes and what can be done about it?

The long version: I have a time-dependent advection-diffusion-reaction problem in one dimension which I need to be able to:

1) simulate
2) linearize
3) compute a steady-state mapping for a varying parameter.

I first tried out the simplest thing I could think of (MOL with a first-order finite-difference scheme for spatial discretization) which turned out to be unstable. After some study, I ended up with implementing a first order upwind finite volume method. This model is stable and I can perform simulations. However, I run into problems with linearization. Since the advective flow can switch sign in the problem, it was necessary to introduce logic for the upwind-scheme. This in turn cause the Jacobian to become discontinuous at the steady-state solution. Is this a common problem and how do I solve it?

One of the reasons I need to linearize is that I want to use continuation in order to compute the steady-state mapping between some states and a control-parameter. Due to the discontinuity, the nullspace of the Jacobian get too many dimensions which cause my continuation algorithm to fail...

I'm now thinking about abandoning the FVM-method and try some collocation method for the spatial discretization. Do you think that is a feasible path?

/Best regards

Ps. I'm rather new to CFD so please, If you know any good litterature for advective-diffusive-reactive problems, I would greatly appreciate tips.