Plane channel driven by a uniform body force: u(y) = G/(2mu) y(1-y), v = 0,
p exactly constant. Convection vanishes identically, so this isolates
diffusion, the half-cell wall treatment and the pressure coupling — and two
of the three answers are exact zeros, which no benchmark comparison offers.
The ends are clamped to the profile the DISCRETISATION prefers — the 1-D
tridiagonal with half-cell wall closures, solved directly in the test —
rather than to the continuous parabola. That makes (u_hat, 0, const) an
exact fixed point of the 2-D discretisation, and the solver must sit on it:
|u - u_hat| ~ 1e-10, max |v| ~ 1e-10, p spread ~ 8e-10 (16^2)
A first version clamped the ends to the continuous parabola instead; the
O(h^2) incompatibility between that profile and the discrete one drove a
weak secondary flow near the ends (max |v| = 1.3e-3) — a property of the
mismatched boundary data, not of the solver, recorded in the test docs so
nobody rediscovers it as a bug.
The wall treatment's own truncation is measured in isolation as
|u_hat - parabola|: 3.906e-3 at 16, 9.766e-4 at 32 — refinement ratio
exactly 4.00, second order, in closed form c h^2 / 4.
Co-Authored-By: Claude Fable 5 <[email protected]>
