[mesa-users] Quality of mass conservation

[Ehsan Moravveji]

Dear Francis,
Thanks for your reply. The finite-volum concept and flux conservation concepts that you put forward fully convinced me.
Now, I understand why it is not so in e.g. Eqs. (7) and (8) of the first instrument paper.

Now, I’m going to start a nice week ;-)

Cheers
Ehsan.


On Oct 27, 2014, at 7:45 AM, Francis Timmes <fxt44 at mac.com> wrote:

> hi ehsan,
> 
> let’s make this simpler. equation 4 of paxton et al 2011,
> from which equation 5 is derived, essentially says
> 
> density_k = mass_k / volume_k
> 
> density_k and mass_k are cell averaged quantities (intensive)
> while volume_k depends on the radii at the cell boundaries (extensive).
> 
> each cell in mesa has some variables that are mass-averaged (intensive)
> and others that are defined at the outer face of a cell (extensive). 
> this way of defining the variables is a consequence of the finite volume, 
> flux conservation formulation of the equations. 
> 
> as the algebraic equality above shows, its perfectly ok to mix 
> intensive and extensive variables for flux conserving quantities.
> your plot supports this notion.
> 
> fxt
> 
> 
> 
> 
> 
>> On Oct 26, 2014, at 2:22 PM, Ehsan Moravveji <e.moravveji at gmail.com> wrote:
>> 
>> Hi everybody,
>> 
>> I’ve been looking at star/private/hydro_eqns.f, and the subroutine do1_density_eqn to figure out how the equality in Eq. (5) in the first instrument paper is fulfilled.
>> So, I’m going to present the first few lines of this subroutine in a jumbled way:
>> 
>> the residual of both sides of the equality are stored in 
>>            equ(equR, k) = lnR(k) - res/3d0
>> where 
>>            lnR(k) = log(s% r(k))
>> and 
>>            res = log_cr(rp13 + dr3)
>> since “lnR" is a cell-edge quantity, then “res" must be also evaluated at cell face, if the comparison is going to be fair.
>> However, one of the ingredients of “res” is a cell-averaged quantity:
>> 
>>            dm = s% dm(k)
>>            rho = s% rho(k)
>>            ! dm/rho is cell volume
>>            dr3 = (dm/rho)/(pi4/3)
>> 
>> Indeed the cell volume is an extensive quantity, but it seems to me it is evaluated at cell center, due to the definition of s% dm and s% rho.
>> I would be grateful is one of the black belts explain to me why this is an allowed operation.
>> 
>> Cheers
>> Ehsan.
>> 
>> PS: B.t.w., for a main-sequence 5 Msun model, the equality in Eq. (5) is fulfilled up to ~ 10 decimals, as you can find out in the attached plot. 
>> Delta = log10((LHS-RHS)/LHS).
>> <equR.png>
>> 
>> 
>> 
>> ------------------------------------------------------------------------------
>> _______________________________________________
>> mesa-users mailing list
>> mesa-users at lists.sourceforge.net
>> https://lists.sourceforge.net/lists/listinfo/mesa-users
>