WASHES · REFERENCE GALLERY · THE STABILITY BOUND
The CFL condition
Courant, Friedrichs & Lewy, 1928. Mathematische Annalen 100, 32–74. The 1928 theorem behind the negative-pigment streaks in Washes pre-v0.69.
An explicit scheme computes each new value from a fixed stencil of old values — its numerical domain of dependence grows by one cell per step. But the true solution at that point depends on data along the physical characteristic, which travels u·Δt per step. The CFL theorem: if the characteristic outruns the stencil, the scheme is computing the answer from data that cannot contain it, and no choice of weights can save it. For donor-cell advection that means |u|·Δt ≤ Δx.
The diagram below shows the space-time cone of grid points an upwind scheme consults (shaded) and the physical characteristic (the bold line) for the chosen Courant number c = u·Δt/Δx. The strip underneath runs live 1-D upwind advection at that same c. The moment the characteristic's foot leaves the cone, the strip grows oscillations and negative pigment.
Space runs horizontally, time upward. The shaded cone is every grid value the scheme can possibly consult when computing the top node. The bold line is where the answer actually comes from.
Try this
1. Sit at c = 1.00 exactly. Upwind becomes the exact shift operator — the bump translates without any smearing at all. The scheme is perfect precisely at the edge of legality, and only there.
2. Back off to c = 0.70. Stable, but the bump erodes: that's numerical diffusion, the price of the safety margin. Stability and accuracy pull in opposite directions for explicit schemes.
3. Step to c = 1.10 and watch the trailing edge. The first thing to appear is a small undershoot below zero — negative pigment — which then feeds back and grows exponentially. This is the cross artifact's signature in one dimension, and the diagram above shows why: the characteristic's foot now lies one node outside everything the scheme can see.