smoothing-spectrum
At its defaults it draws what one weighted jacobi sweep does to each mode, at ω = 0.67. A curve of the damping factor against mode frequency, falling from one at the left through zero and towards minus one at the right. The right-hand half of the plot is shaded, and two horizontal lines mark the worst damping inside it.
smoothing-spectrum is one function in lib/figures/multigrid.js —
multigrid — the smoother, the coarse grid, and the rate that does not move. 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.
A curve of the damping factor against mode frequency, falling from one at the left through zero and towards minus one at the right. The right-hand half of the plot is shaded, and two horizontal lines mark the worst damping inside it.
omega: 0.6667
The arguments are the ones A rate that does not notice the size 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.
A curve of the damping factor against mode frequency, falling from one at the left through zero and towards minus one at the right. The right-hand half of the plot is shaded, and two horizontal lines mark the worst damping inside it.
omega: 1
The arguments are the ones The error smoothing cannot reach 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.
A curve of the damping factor against mode frequency, falling from one at the left through zero and towards minus one at the right. The right-hand half of the plot is shaded, and two horizontal lines mark the worst damping inside it.
omega: 0.6666666666666666
The arguments are the ones The stencil that is not symmetric 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.
A curve of the damping factor against mode frequency, falling from one at the left through zero and towards minus one at the right. The right-hand half of the plot is shaded, and two horizontal lines mark the worst damping inside it.
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.
14 distinct claims across 4 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.
mode 1 is damped by the factor its frequency predicts agree — asserted 12 times
the smoothest mode survives whatever ω is
the smoothing factor, scanned and in closed form agree
Against the rule
The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.
Across the library: the rule bites on 52
of 99 generators —
37 print a residual and
15 are exempt with a published reason;
47 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.
A rate that does not notice the size
The V-cycle reduces the residual by a factor of ten a cycle at fifteen points and at a hundred and twenty-seven. Jacobi on the same four problems goes from 0.981 to 0.9978, climbing towards one. One of those is a constant and the other is an exponent, and that is the whole distinction the field turns on.
Iterating, instead of factorisingSmoothing a whole line at once
Solve every grid line in the strong direction exactly rather than sweeping over it, and the smoothing factor goes from 0.9993 back to 0.3340 — which is the one-dimensional answer, on a problem that is not one-dimensional. The repair replaces one ε in the closed form by a one.
Iterating, instead of factorisingThe error smoothing cannot reach
One weighted Jacobi sweep multiplies every mode of the error by a number, and the number is a sine. Half the modes are cut by three or better, and the other half come back at 0.999 — which is not a failure of the method but the fact the whole of multigrid is built on.
Iterating, instead of factorisingThe same problem on a coarser grid
Restriction, the coarse operator and interpolation are three matrices with nine distinct entries between them. Two of the three are each other's transpose, and their product with the fine operator is the coarse discretisation exactly — not approximately, entry for entry, at every level.
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.