In CDKLM the cross flux term, huv, is evaluated upstream, "for consistency". It is not obvious what this consistency represents, and experimentation in commit d92b75b, branch havahol/cdklm_central_upstream, shows that second-order convergence is maintained also by using a central-upwind flux similar to the other flux terms.
We should look closer into which upwind formulation should be used for the cross flux terms.
The experiment above is based on PR #154