[FLASH-USERS] using the split PPM hydro solver in 3D spherical polar coordinates

[Klaus Weide]

On Tue, 23 Dec 2014, Rodrigo Fernandez wrote:

> The point about the reconstruction is a separate one. For consistency with
> the Eulerian formulation of the hydrodynamic equations in conservative
> form, the PPM polynomial that yields the first guess at the zone interface
> value should assume that the independent coordinate $\xi$ is the volume.
> Otherwise the interface value does not arise out of volume-average values.
> 
> Colella & Woodward (1984) mention this explicitly in their paper (third
> paragraph of section 3) when discussing the Eulerian formulation of PPM. It
> just so happens that when using cartesian coordinates, the volume is
> proportional to the coordinate.
> 
> The subroutine coeff.F90 takes the coordinate increments (dx) as input to
> compute the coefficients.
> 
> Perhaps someone already thought about this a long time ago and I'm
> inventing the wheel here...

Rodrigo,
Thanks for clarifying.

Just as another aspect to consider: the PPM algorithm as used already
forgoes the benefits of having a conservative method of reconstruction - 
at least partially, for the variables other than density. That is because
the conservative reconstruction is applied to variables in non-
conservative form (velocity instead of momentum density, pressure instead 
of energy density, mass fractions instead of partial densities).

So if the conservative nature of some variables can be so cavalierly*
discarded even in Cartesian geometry, then that makes it look more okay to
ignore conservatism in additional ways for curvilinear coordinates...

Feel free to point out errors in my analogy.


* I don't know whether there are positive reasons for doing reconstruction 
this way, haven't gone back to the scriptures for this but am willing to 
be educated.

Klaus