original SDC implementation Jacobian is not correct

[zingale]

The analytic Jacobian for the original SDC (enabled by setting the SDC preprocessor) is not correct. The Jacobian elements should be derivatives with respect to the conserved variables. The proper way to do this is to construct a second transformation Jacobian, d(rho, rho X, rho E)/d(rho, T, X) and use this to transform the derivatives that come from the burner. This is actually quite easy, and is implemented in the new SDC implementation in the sdc branch of Castro.

With the analytic Jacobian enabled (jacobian=1), this original SDC method surprisingly runs, but very slowly.

[zingale]

At the moment, our state vector that the ODE integration sees is (rho X, rho E, rho e). We get the Jacobian from the net in terms of rho, T, X. I think in order to do the transformation properly, we should integrate only one of (rho E) or (rho e), and add rho to the ODE integration vector. Then we can do the same transformation as we are doing in the new SDC stuff.

[zingale]

It looks like the function that needs to be modified is jac_to_vode in vode_type_simplified_sdc.F90