Error against the angle between the flow and the grid, at Pe = 3.13

At its defaults it draws error against the angle between the flow and the grid, at pe = 3.13. Three curves against the flow angle on a logarithmic vertical axis. The tuned scheme is exact at zero — 2.4·10⁻¹⁷ — and is 0.00794 five degrees later. Central differencing and upwinding start at 0.517 and 0.136 and fall as the angle grows, because the exact solution's own size in the interior falls with it.

angle-cliff is one function in lib/figures/supg2d.js — the tuned diffusion in two dimensions — exact along one axis, and the direction it does not go. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

Three curves against the flow angle on a logarithmic vertical axis. The tuned scheme is exact at zero — 2.4·10⁻¹⁷ — and is 0.00794 five degrees later. Central differencing and upwinding start at 0.517 and 0.136 and fall as the angle grows, because the exact solution's own size in the interior falls with it.

Three curves against the flow angle on a logarithmic vertical axis. The tuned scheme is exact at zero — 2.4·10⁻¹⁷ — and is 0.00794 five degrees later. Central differencing and upwinding start at 0.517 and 0.136 and fall as the angle grows, because the exact solution's own size in the interior falls with it.

Three curves against the flow angle on a logarithmic vertical axis. The tuned scheme is exact at zero — 1.7·10⁻¹⁶ — and is 5.86·10⁻⁴ five degrees later. Central differencing and upwinding start at 0.0556 and 0.157 and fall as the angle grows, because the exact solution's own size in the interior falls with it.

Three curves against the flow angle on a logarithmic vertical axis. The tuned scheme is exact at zero — 1.9·10⁻¹⁷ — and is 0.0141 five degrees later. Central differencing and upwinding start at 0.734 and 0.0741 and fall as the angle grows, because the exact solution's own size in the interior falls with it.

Three curves against the flow angle on a logarithmic vertical axis. The tuned scheme is exact at zero — 2.6·10⁻¹⁷ — and is 0.00474 five degrees later. Central differencing and upwinding start at 0.367 and 0.178 and fall as the angle grows, because the exact solution's own size in the interior falls with it.

Three curves against the flow angle on a logarithmic vertical axis. The tuned scheme is exact at zero — 3.5·10⁻¹⁷ — and is 0.00303 five degrees later. Central differencing and upwinding start at 0.263 and 0.198 and fall as the angle grows, because the exact solution's own size in the interior falls with it.

Every figure above checked its own claims on the way to being drawn, and a claim that failed would have stopped the picture rather than shipped a wrong one. Those checks used to leave no trace at all: a passing one returned true and the only evidence the figure had checked anything was that nothing crashed. The list below is what they actually said, collected by running this generator with an observer installed — not a description of what it is believed to check.

8 distinct claims across 6 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.

It draws a decomposition and prints its residual. It calls solve2d, and every figure above carries the badge — which residualcheck verifies by looking for it in the emitted SVG rather than by finding the call that builds one. A badge that is constructed and then left out of the body is the failure that check exists for.

Across the library: the rule bites on 217 of 397 generators — 199 print a residual and 18 are exempt with a published reason; 180 factorise nothing. Read from lib/residual-rule.js, which is the same body the gate enforces from, and the gate's last check fails the build if this page and it disagree about any generator.