WASHES · REFERENCE GALLERY · THE GRID
Realistic Animation of Liquids
Foster & Metaxas, 1996. Graphical Models and Image Processing 58(5), 471–483. The first Navier–Stokes solver in graphics — and the staggered MAC grid Washes inherits through Curtis et al.
On a MAC (marker-and-cell) grid, the unknowns don't live in the same place: pressure sits at cell centers, the horizontal velocity u on vertical faces, the vertical velocity v on horizontal faces. The payoff for that bookkeeping is shown below. The cells are shaded by a pressure field; the green arrows are the pressure-gradient forces the solver actually computes from it.
Set the pressure to a checkerboard and switch between discretizations. The staggered grid differences adjacent cells across each face and sees the checkerboard at full strength. A collocated grid, differencing across two cells with a central difference, returns exactly zero everywhere — the mode is invisible to the solver, so nothing ever removes it, and it contaminates the pressure solve as decoupled checkerboard noise. Staggering eliminates the null mode by construction.
Try this
1. Checkerboard + collocated. Every sampled gradient is exactly zero: p[i+1] and p[i−1] are always equal on a checkerboard, so the central difference cancels perfectly. The most violently oscillating field possible produces no restoring force at all.
2. Checkerboard + staggered MAC. Now each face differences immediate neighbors, and the forces are the largest the grid can represent. The mode the collocated grid couldn't see is the one the MAC grid sees best.
3. Smooth bump under both discretizations. For well-resolved fields they broadly agree — the arrows point away from the high-pressure center either way. The grids differ only on the highest-frequency mode, which is exactly where it matters: that's the mode round-off and source terms continually inject.