are we filling the flux registers correctly for SDC?

[zingale]

Do we need to weight the fluxes added to the flux registers by the dt between time nodes? Or perhaps using the quadrature rule?

[zingale]

actually.. I think it is the quadrature rule weights. And we should only add to the flux register for the last iteration.

[zingale]

it's a bit more tricky...

since we only compute the advection term once, for iteration = 0, we need to store the flux in the flux register then. The remainder of the time nodes need to be added to the flux register on the last iteration, all with the proper weights. So the flux register needs to persist over the entire iteration structure.

[zingale]

confusion no. 2:

if we are doing 4 iterations, then we have the following:

iteration
0 : store node 0
1 : update the state using the integral over iteration 0
2 : update the state using the integral over iteration 1
3 : update the state using the integral over iteration 2

so do we want to use the fluxes from iteration 3 or 2 for the flux registers?