How to check whether a numerical wave is a shock

Identify the two physical problems being compared

One spherical gas calculation cools under gravity and receives a central thermal pulse during eighteen to nineteen million years. The other starts from the same atmosphere and only cools. The thermal energy is five percent of the eighteen-million-year total input; power, schedule and spatial kernel stay fixed.

Spatial comparison uses 2048 and 4096 radial cells. The timestep comparison halves the CFL coefficient from 0.35 to 0.175 on the same 4096 grid. This coefficient limits how far the fastest signal travels across the grid in one step. Every case runs from its initial state to sixty million years.

Each spherical record retains twelve predeclared complete fields, 334 integral rows and 81 candidate-diagnostic frames. Complete fields are retained at the twelve declared times. Conservation and physical-domain checks cover every accepted step.

Reading: Sandbox Physics

A cell mean is not a centre-point sample

A finite-volume cell represents a shell average. Combining two fine cells requires their actual shell volumes. A simple arithmetic mean gives the larger outer shell too little weight.

Reconstruct density, radial velocity and pressure in each physical calculation, then subtract matching cooling. Project the fine-grid difference onto coarse volumes before comparing it. The independent reader script streams through the shells to check the same volume integrals.

\overline f_C=\frac{1}{V_C}\sum_{j\subset C}V_j\overline f_j,\qquad V_j=\frac{4\pi}{3}\left(r_{j+1}^3-r_j^3\right)

Density, velocity and pressure are tested separately, preserving their distinct units.

Read the denominator before the percentage

The error divides the volume integral of the absolute discrepancy by the actual pulse-minus-cooling signal. Each variable and saved time is tested separately. Before heat release, the first three signals are exactly zero; those cases use an absolute-error check.

e_f(t)=\frac{\sum_C V_C\left|\delta f_C^{\rm coarse}-\mathcal P\delta f^{\rm fine}_C\right|}{\sum_C V_C\left|\mathcal P\delta f^{\rm fine}_C\right|}

Projection uses shell volumes. The original spatial limit is five percent and the half-timestep limit one percent.

Actual comparison	Largest relative field error	Result
512 to 1024	0.178	Spatial fail
1024 to 2048	0.0794	Spatial fail
2048 to 4096	0.0332	Spatial pass
Half timestep, original coarse projection	$4.68727\times10^{-6}$	Time pass
Half timestep, additional native 4096 comparison	$4.85967\times10^{-6}$	Time pass

Reading: Sandbox Physics

Associate the same propagation branch

Several compression bands may coexist. Selecting the strongest feature independently in every frame can jump from a fading pulse to a growing one. Differencing those positions invents a speed that no single front travelled.

The diagnostic retains all bracketed negative-gradient minima in its window. Adjacent quarter-million-year frames require unique bidirectional links using actual upstream flow, sound speed and the predeclared weak-shock Mach interval. Ambiguous links, a selected-branch switch or a stronger truncated boundary shoulder make the candidate speed unavailable.

This establishes local kinematic consistency only. It neither classifies a minimum as a shock nor forces distant assessment times into one trajectory.

Reading: Sandbox Physics

Compare excess entropy with its matching floor

Heating and cooling themselves change entropy. An advected source-corrected label separates their declared contributions from the excess diagnostic. Candidate entropy is then compared with the largest acoustic-control, matching-cooling, spatial or timestep floor.

Matching cooling dominates all four final floors. At thirty million years, excess entropy is about 1.21 times ten to the minus ninth, versus a floor about 5.67 times ten to the minus eighth. Their ratio is 0.02134; the required ratio is greater than ten. The small negative value at twenty-six million years lies below the floor and is retained as unresolved numerical excess.

Time / million years	Excess entropy / original floor	Original RH jump relative error
26	−0.126	0.709
30	0.0213	0.367
34	0.0247	0.458
38	0.0277	0.304

Reading: Sandbox Physics

Admit the gentle wave and retain the shock failure

A shock must satisfy all the original conditions: a location beyond the source and stencil, compression, a Mach number above one and at most 1.3, consistent density/pressure jumps, separation from sonic numerical uncertainty, and entropy clearly above the matching floor. At least two consecutive assessment times must pass.

All four times fail jump, sonic-separation and entropy conditions. Refined pulse-gradient half-height widths are about 0.38 to 0.61 kiloparsecs, spanning about 19 to 65 local cells. These describe a finite waveform; they are not fitted discontinuity thicknesses.

A separate uniform ideal-gas RH reference passed its own tests. It supplies evidence for its own prescribed uniform-gas problem. The AGN pulse keeps its separate shock assessment.

Reading: Sandbox Physics · Forman et al. (2007)

Replay the two field norms locally

Download the full packet, native refinement ZIP, manifest, convergence record and independent script. ZIP members preserve the original NPZ bytes. The packet keeps all 4096 radii and twelve actual timestamps without downsampling or interpolation.

The replay gives spatial maximum 0.03321494 and original coarse-projected timestep maximum about 4.687 parts per million. Both maxima occur in the final-frame velocity column, and early zero-signal errors are zero. This recomputes field norms; the separate saved results describe per-step conservation, benchmarks and shock identification.

            python3.12 -m venv .venv
. .venv/bin/activate
python -m pip install numpy==2.3.5
python reproduce-wave-refinement.py --materials . --output local-wave-check.json

Reading: Sandbox Physics

Download the independent scientific materials

The full packet and native refinement archive retain every saved radial cell and timestamp. The independent script checks data identities and recomputes the actual field norms.

Exact byte, SHA256 and license manifest

LICENSE-code.txt
1,362 bytes · Original calculation/documentation with attribution; see license notices
96630e05864696f836e37b8b3c5abf777a90485dbd8737d21eadd44ce0235613
LICENSE-data.txt
1,054 bytes · Original calculation/documentation with attribution; see license notices
67ae54be72c18d72a70cb63400f912e5e31d43c36938aef6504bd89543e20223
gentle-wave-convergence.json
47,943 bytes · Original simulated numerical outputs; attribution grant LICENSE-data.txt
3b55a6e37a37c5debe3b56843d9f7b7fd8c713b210a5483199fdea1305748e1e
reproduce-wave-refinement.py
6,633 bytes · MIT; LICENSE-code.txt
4ee1c692c5c1243cca1eeb7422d996b64eb0ae26e5233108f5084611e9319d2b
validate-records.py
3,382 bytes · MIT; LICENSE-code.txt
f89262e63c4f4352ef1ff3f9a19f396045acfddd9e7bee4a921c558a633a1dd6
wave-refinement-expected.json
4,574 bytes · Original simulated numerical outputs; attribution grant LICENSE-data.txt
15e692ba0b761d78eac6e59bab4f25689b82a6b4edad42652b1a287617d0148e
wave-refinement-inputs.zip
4,542,941 bytes · Original simulated numerical outputs; attribution grant LICENSE-data.txt
cea357cad48784dcd8e0ce1283c2ce69e5cba6cc34023e7f519433859e9d30a2
wave-refinement-manifest.json
3,753 bytes · Original simulated numerical outputs; attribution grant LICENSE-data.txt
bdce85295190991805578c19b26cc772b381439b40587c6c43320fd28abeee4a
accepted-records.json.gz
2,869,078 bytes · Original full-precision simulated output; attribution grant LICENSE-data.txt
2a81501bed03e97dbf5ed6df9ef96b6e452669eed7bb628d98e530c35dc8c236

Try the observation

Read the complete waveform

At thirty million years, move a probe through several compression and rarefaction regions.
Follow complete profiles through time and check that a proposed front keeps its identity.
Distinguish visibility of a compression wave from the shock conditions in the explanation.

Compare a different shock problem

Read that Lab’s own initial conditions and assumptions.
Compare its jump relations without combining its source energy with this gentle pulse.

Exercises and explained answers

Why refine space if the half-timestep error is already small?

The two errors arise from different discretizations. Earlier timestep errors were small while spatial waveform errors exceeded ten percent. Finite spatial resolution still affects the waveform after the timestep is reduced.

Why can connecting the steepest gradient in every frame invent a supersonic speed?

The selected maximum can switch between different pulses. The difference includes the separation between two branches; the inferred trajectory changes identity.

Primary sources and calculation notes

Forman et al. (2007), Filaments, bubbles, and weak shocks in the gaseous atmosphere of M87
Density and temperature jumps support the weak-shock interpretation in a different system, M87. Used as an observational contrast, not a calibration of the gentle pulse.
Sandbox Physics, frozen gentle-source numerical evidence (2026)
Actual 512/1024/2048/4096 refinement history, original shock criteria and matching controls. The source power and scientific gates were kept fixed; the final scope remains no resolved AGN shock.