NOTES

Very close to having the adjoint solver puzzled out
Need to finish the exact solution code

Explicit Solver:
Does not converge if mref < 1.0,
velocity is allowed to go negative which could be the problem.
Some problems with coarse grid convergence...
could just be a van Leer FVS dissipation problem, but need to analyze

Implicit Solver:
Dividing through by cell_vol seems to have fixed CFL stability problem,
still need to revisit time step definition.
Can ramp/jump from low cfl to over 100 now.

Can run 2nd order with a smooth limiter ( i.e., not minmod, superbee ),
but convergence will stall depending on options...
might need to introduce limiter freezing?

Need to nondimensionalize, the matrix structure for the flux Jacobians is:

Row 1: O(1)
Row 2: O(100)
Row 3: O(100000)

This is not good for the numerics... will most likely nondim. on STP conditions.

TODO:
Need to check a linear starting solution.
Need to create cell_vol vector rather than constantly recalculating it
Need to create Roe LHS... although Q1D does run with van Leer LHS and Roe RHS
Need to test linear extrapolation LHS fix

Still need to investigate what the cell metric term is for this formulation...
should it include cell area or not?

Update April 2nd, 2012
Need to make vector of variable limiters and
introduce these to the LHS so the adjoint works.

AND/OR
Make Jacobians be wrt primitive variables and then convert to conserved...?
Need to check math on this.

Update April 4th, 2012
Supersonic outflow with totally subsonic inflow initial conditions will not
converge with kappa = -1.0 or 0.333333333333!
mref = 1.5 is a good starting constant initial condition for 1st order lhs with
a 1st or 2nd order rhs

Update April 11th, 2012
After making a linear ramp for the initial conditions,
convergence is much improved.
Have finally been able to test convergence rates on the fully isentropic case.
Ramping CFL from 1 to 2000 over 10 iterations shows that the 2nd order LHS
outperforms the 1st order LHS, although 27 vs 46 iterations is essentially
instantaneous Newton convergence.
Ramping CFL from 1 to 6000 over 10 iterations causes the 1st order LHS to
become unstable, while the 2nd order LHS converges in 22 iterations.

Update June 27th, 2012
Need to freeze limiters for the adjoint