problem with fluxes at interfaces AMRCLAW

[Eva Tuholi]

Hello,
I am using AMRCLAW 4.3 and I am in 1- Dimension. My = 1.
I have record the fluxes at each interface. I can see by the values
that fm(i+1) wich would be the left flux at interface i+1/2 (and hence
the flux affecting cell i), is different that fp(i+1) which should be
the right flux at interface i+1/2.
I have record the fluxes that routine stepgrid.f returns. I was
thinking those fluxes should be the same for a conservative scheme .
is there anything I missed?
thanks

[Kyle Mandli]

If I understand correctly, you are talking about the fluxes leaving
cell i+1?  Something like

------------+-----------------------+-----------------------+----------------
i-1 i i+1 i+2
<------ fm(i+1)
fp(i+1) ----->

Hopefully that showed up in the email correctly. If that's the right
diagram, there's no reason to expect those fluxes to be the same. The
values of ql(i+1) and qr(i+1) should be the same but that's an
algorithmic result, not due to conservation.

I think I may be missing what exactly it is what you are trying to ask
so hopefully the diagram can help us.

Kyle

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[Eva Tuholi]

Maybe you will more than one e-mail, but i am trying to fix the
picture . please forgive me for the number of e- mails
int. i-1/2 int. i+1/2
n+1
-----o----------][-----------o----------][---------o---------][--
i-1 i i+1
I I
I -------> I <-------
I fp(i+1) I fm(i+1)
n---o----------][-----------o----------][--------- o---------][--
i-1 i i+1


When we do flux differencing formula
q(i)^(n+1) = q(i)^n - k/h * [F(i+1/2)-F(i-1/2)]
I was thinking that
fm(i+1) = F(i+1/2)
and for
q(i+1)^(n+1) = q(i+1)^n - k/h * [F(i+3/2)-F(i+1/2)]
fp(i+1) = F(i+1/2)
so I was expecting fm(i+1) = fp (i+1)

[Eva Tuholi]

[Kyle Mandli]

How are you trying to calculate fp and fm?

In Clawpack the values calculated from the Riemann solver are apdq and
amdq, the fluctuations coming from grid cell boundaries which resemble
your fp and fm but are something slightly different. The evaluations
of the flux function itself in the states to the left and right of a
grid cell boundary is also expected to be different and is why we
solve the Riemann problem. The numerical flux F^n_{1-1/2} it looks
like you are talking about can be calculated via

F^n_{i-1/2} = f(q_i) - apdq_{i-1/2} = f(q_{i-1}) - amdq_{i-1/2}

where f(q_i) is the flux function evaluated with state q_i and apdq
and amdq are the fluctuations moving from the grid cell boundary
located at i-1/2. In the case of conservation laws and a conservative
numerical method the value of F^n_{i-1/2} is unique for each grid cell
boundary.

Kyle