wrong-problem
At its defaults it draws the same three schemes on a problem with no layer in it. Worst nodal error against the grid size, both axes logarithmic, for a manufactured smooth solution on the identical operator at ε = 0.005. Adding no diffusion gives 0.0016, 4·10⁻⁴, 10·10⁻⁵, falling by four at each refinement. The tuned diffusion gives 0.067, 0.022, 0.0061 — 42 times worse at the coarsest grid, and falling more slowly.
wrong-problem is one function in lib/figures/supg.js —
tuned diffusion — exact at the nodes, on the problem it was derived from. 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.
At its defaults
Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.
Worst nodal error against the grid size, both axes logarithmic, for a manufactured smooth solution on the identical operator at ε = 0.005. Adding no diffusion gives 0.0016, 4·10⁻⁴, 10·10⁻⁵, falling by four at each refinement. The tuned diffusion gives 0.067, 0.022, 0.0061 — 42 times worse at the coarsest grid, and falling more slowly.
eps: 0.005
The arguments are the ones An answer that is known passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
Worst nodal error against the grid size, both axes logarithmic, for a manufactured smooth solution on the identical operator at ε = 0.005. Adding no diffusion gives 0.0016, 4·10⁻⁴, 10·10⁻⁵, falling by four at each refinement. The tuned diffusion gives 0.067, 0.022, 0.0061 — 42 times worse at the coarsest grid, and falling more slowly.
eps: 0.02
The arguments are the ones The exact answer to a nearby problem passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
Worst nodal error against the grid size, both axes logarithmic, for a manufactured smooth solution on the identical operator at ε = 0.02. Adding no diffusion gives 0.0016, 3.9·10⁻⁴, 9.7·10⁻⁵, falling by four at each refinement. The tuned diffusion gives 0.024, 0.0061, 0.0015 — 15 times worse at the coarsest grid, and falling more slowly.
What it checked while drawing
Every figure above asserted its own claims on the way to being drawn, and a claim that failed
would have failed the build rather than drawn a wrong picture. Those assertions 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 3 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.
the tuned scheme is worse than plain central differencing at n = 31 — asserted 3 times
and converges more slowly at n = 63 — asserted 2 times
a diffusion the comparison is drawn at
enough sizes to fit a rate
LU is for square matrices
Against the rule
It draws a decomposition and prints its residual. It calls
solveManufactured,
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 66
of 131 generators —
51 print a residual and
15 are exempt with a published reason;
65 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.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
An answer that is known
Almost every demonstration of numerical error estimates the error by computing the same thing more carefully. The Hilbert matrix does not need that: its inverse is a closed form in integers, so the true answer is available exactly and the error is measured rather than approximated.
Iterating, instead of factorisingThe diffusion that makes the answer exact
Upwinding adds h/2 of artificial diffusion. Central differencing adds none. Add ε·ξ·Pe with ξ = coth(Pe) − 1/Pe and the computed solution is the exact one at every grid point, to 2.4·10⁻¹⁷ — at every Péclet number, on the problem it was derived from and on no other.
Two errors, and whose fault they areThe exact answer to a nearby problem
A good algorithm does not give an approximate answer to your problem. It gives the exact answer to a problem very close to yours — and once that is the definition, a wrong result has two possible authors and they can be measured apart.
Iterating, instead of factorisingThe stencil that is not symmetric
Past a cell Péclet number of exactly one — measured by bisection at 1.0000000000000002 — the central-difference solution of a convection–diffusion problem oscillates from point to point and leaves the interval the equation guarantees, at 16 of 31 grid points. It is the exact solution of its own linear system, to 4.6·10⁻¹⁸. No solver was involved.