complex-symbol
At its defaults it draws the damping of each fourier mode, ε = 0.005, ω = 0.6667. The modulus of one weighted Jacobi sweep's effect on each grid mode, against the frequency. On the operator with no convection in it the effect is a real number and the oscillatory half is damped to ⅓ at ω = 2/3. With convection the symbol is complex, its imaginary part does not depend on ω, and the worst damping over the oscillatory half is 1.0937.
complex-symbol is one function in lib/figures/convection.js —
convection — the stencil that is not symmetric, and the fourier arguments that assumed it was. 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.
The modulus of one weighted Jacobi sweep's effect on each grid mode, against the frequency. On the operator with no convection in it the effect is a real number and the oscillatory half is damped to ⅓ at ω = 2/3. With convection the symbol is complex, its imaginary part does not depend on ω, and the worst damping over the oscillatory half is 1.0937.
omega: 0.6666666666666666
The arguments are the ones A direction the smoother cannot see 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.
The modulus of one weighted Jacobi sweep's effect on each grid mode, against the frequency. On the operator with no convection in it the effect is a real number and the oscillatory half is damped to ⅓ at ω = 2/3. With convection the symbol is complex, its imaginary part does not depend on ω, and the worst damping over the oscillatory half is 1.0937.
omega: 0.35
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.
The modulus of one weighted Jacobi sweep's effect on each grid mode, against the frequency. On the operator with no convection in it the effect is a real number and the oscillatory half is damped to ⅓ at ω = 2/3. With convection the symbol is complex, its imaginary part does not depend on ω, and the worst damping over the oscillatory half is 0.8495.
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.
6 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.
a diffusion coefficient inside the range drawn
a relaxation parameter inside the range
because the operator is far from symmetric
no relaxation parameter reaches the symmetric operator's smoothing factor
the symbol has an imaginary part
the symmetric smoothing factor is exactly one third at ω = 2/3
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 direction the smoother cannot see
Give the Laplacian a strong direction and multigrid stops working — from 0.2016 a cycle to 0.9565 — with every component unchanged and the condition number identical to twelve digits. The problem did not get harder. The link between the method's two halves broke.
Iterating, instead of factorisingA rate that is known in advance
On the model problem, Jacobi contracts by cos(π/(n+1)) per step, Gauss–Seidel by its square, and optimally relaxed SOR by a number given in closed form. Three rates, all known before anything runs, and all measurable against what runs.
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.