2-D prototype, round 6: explicit advection with kappa storage on cut cells (small-cell problem).
Flow: solid-body rotation psi = r^2/2 in the annulus a < r < R (a = 0.4, R = 1), both circles
immersed. Face fluxes are EXACT integrals of the stream function over the OPEN part of each face,
so the discrete aperture-weighted divergence vanishes in every cut cell (as for flow's projected
MAC flux). A Gaussian blob next to the outer wall is carried once round (T = 2 pi); the exact
answer is the initial field. dt = CFL * h / max|u| with the FULL-cell CFL 0.5.
Schemes (first-order upwind fluxes throughout, to isolate the storage / small-cell question):
unit : (c^{n+1}-c^n) h^2/dt = -sum F c_up (flow today: conserves sum c, not sum kappa c)
kappa : kappa h^2 (c^{n+1}-c^n)/dt = -sum F c_up (explicit; unstable for small kappa?)
mb : kappa storage; faces touching a cell with kappa < 1/2 use IMPLICIT upwind values, all
other faces explicit (May & Berger-style explicit-implicit split; conservative because
each face flux is shared by its two cells); one sparse solve per step
impl : kappa storage, all faces implicit (reference for stability)
Reported: max/min of c (overshoot = instability or max-principle violation), relative drift of
the physical mass sum kappa c, L1 error vs the exact (initial) field over the fluid volume.
1.9.8