Exact answer known1D · pressurizedCandidate

Pollutant spreading in a pipe

Does the model carry and spread a pollutant correctly?

Test A1P-030
Ogata (1961)
Version 1.0.0

Time since the start

Press play, or drag the slider, to watch the exact answer unfold. Models are compared at the end, t = 3,600 s.

What happens

A tracer is released continuously into steady flow. After an hour its front has moved 360 m and smeared out by dispersion, following an exact formula.

Why it matters

Contamination warnings, chlorine decay and sewer water quality all depend on getting both the travel and the spreading right.

A good model must

match the concentration profile within 2 %
conserve the tracer within 0.5 %
show second-order accuracy on finer grids

How models did

In rank order: by each model's percent difference from the right answer, the largest over the test's accuracy checks (lower is better; results within 10 % of each other share a rank). Mass error, timing and NSE are shown beside it, not folded in. Each model runs the same inputs on 4 grids, from 4 m down to 0.5 m cells; the checks are scored on the 0.5 m grid, and each card lists the error on every grid. Compare them in the results viewer →

Rank	Model	Difference	Largest difference in	Mass error	Run time
1	Demo 1D suite example	0.045 %	Concentration profile	<0.01 %	–
2	EPANET 2.2 US EPA	1.9 %	Concentration profile	0.028 %	0.56 s

Demo 1D

Worked example that ships with the suite: plain, well-understood schemes

#1 0.045 % off

Rate the error shrinks on finer grids: 1.5 (90 % range 1.3 to 1.8); the method's expected rate is 1.6 or more

Concentration profile

• 0.045 % off

Tracer conserved

· 0.0071 % error

conservation, reported beside the score

Error by grid (L1 c)

· 4 m: 0.00399 · 2 m: 0.00162 · 1 m: 0.000561 · 0.5 m: 0.000161

reported, not counted · the checks above are scored on the 0.5 m grid

Improves with finer grids

· rate 1.54 (90 % range 1.31 to 1.78)

reported, not counted · the scheme's design order is 2

Scored with method 3.0 · Score card: demo

Run it yourself

demo · version 0.1.0 · runs on any · produced by scripts/run_vertical_slice.py in track-a.yml.

git clone https://github.com/scott-jeffers/benchmark1d2d && cd benchmark1d2d
git checkout 71dc10fc07a9
python -m pip install -e ".[dev,engines]"
python scripts/run_vertical_slice.py

The model files for this run will be published with its next run.

The test itself (inputs, exact answer, checks): A1P-030.zip. Check the score card with benchmark1d2d verify-card: its signature by the suite's public development key shows the card has not changed since it was written, not who wrote it.

EPANET 2.2

EPA EPANET 2.2, the standard water-network solver

#2 1.9 % off

Rate the error shrinks on finer grids: -0.0043 (90 % range -0.0087 to 0.0002); the method's expected rate is 1.6 or more

Run time, finest grid: 0.56 s · AMD EPYC 9V74 80-Core Processor, 2 cores, 15.6 GB, Linux 6.17.0-1022-azure (GitHub Actions (github-hosted, ubuntu24))

Settings: epanet (WNTR 1.5.0, libepanet22); water quality, Lagrangian advection (no dispersion)

Concentration profile

• 1.9 % off

Tracer conserved

· 0.028 % error

conservation, reported beside the score

Error by grid (L1 c)

· 4 m: 0.00671 · 2 m: 0.00675 · 1 m: 0.00676 · 0.5 m: 0.00677

reported, not counted · the checks above are scored on the 0.5 m grid

Improves with finer grids

· rate −0.00 (90 % range −0.01 to 0.00)

reported, not counted · the scheme's design order is 2

Scored with method 3.0 · Score card: EPA EPANET 2.2

Run it yourself

EPA EPANET 2.2 · runs on Linux · produced by scripts/run_network_cases.py in track-a.yml.

git clone https://github.com/scott-jeffers/benchmark1d2d && cd benchmark1d2d
git checkout 71dc10fc07a9
python -m pip install -e ".[dev,engines]"
python scripts/run_network_cases.py

The model files the suite wrote for this run (.zip, 51 KB): the exact inputs the solver read (one folder per grid, where the test has several), with their SHA-256.

The script writes the EPANET input file from the test's inputs; the model files below are that file.

Not run yet: HEC-RAS 7.0, TELEMAC-2D, ANUGA, GeoClaw, LISFLOOD-FP, TUFLOW, MIKE 21, Iber, BASEMENT, Delft3D FM

Run it and add yours →

Error shrinks as the grid gets finer

Average error on each grid. A steeper line means the model converges faster.

Demo 1D · rate 1.7EPANET 2.2 · rate -0.0016

Status of this test

Candidate. To become frozen it needs: maintainer review recorded.

Download inputs (.zip)Check your results in the browser Cite this test View on GitHub

Technical details for reviewers Checks, reference, inputs, sources

Checks and why each limit was chosen

Check	Difference measured as a percentage of	Latest	Why this check
concentration_L1 L1 of c · in the score	already a percentage	–	Concentration L1 error within 2 % of the L1 norm of the Ogata–Banks profile on the finest grid. The front is 8.5 m wide (2√(Dt)) and moves 360 m, so a scheme with numerical diffusion comparable to D (first-order upwind at u·dx/2 = 0.025 m²/s, 2.5×D) visibly smears it and fails.
mass global_mass_error of c · conservation, beside the score	already a percentage	–	Tracer mass in the domain within 0.5 % of the exact integral: a conservative scheme meets it to quadrature error.
convergence observed order of L1 error in c · reported, not counted	the design order, 2	–	Measured from the error on every refinement level, as in ASME V&V 20.

Reference

Exact solution from references.ogata_banks.concentration.

More figures

Exact, t = 0.25 h
Exact, t = 0.5 h
Exact, t = 0.75 h
Exact, t = 1 h
Demo solver, t = 1 h

Exact profile every 15 min; dots are the demo solver (MUSCL advection, central diffusion) on its finest grid, around the front at 1 h.

Promotion checklist

Only frozen tests have fixed inputs and answers. This list is computed from the manifest and the published score cards; the rules are in the method.

✓Data licence verifiedfor candidateanalytic-public
✓All inputs availablefor candidateall committed or fetched by SHA-256
✓Reference (truth) availablefor candidateanalytic generator
✓Every tolerance justifiedfor candidateeach gating criterion has a rationale
✓A published run passesfor frozen2 passing score card(s)
✓Order of accuracy shownfor frozenobserved order meets the minimum
✓Two independent codes passfor frozenpassing codes: EPA EPANET 2.2, demo
✗Maintainer review recordedfor frozenno maintainer review recorded

Inputs

File	Role	Format	SHA-256	Get it
`inputs/params.json`	case_parameters	json	`94ed23813123…`	Download

About this test

Physics

A tracer enters a uniform flow $u = 0.1$ m/s at $x = 0$ with $c = c_0$ from $t = 0$; dispersion $D = 0.01$ m²/s. Ogata & Banks (1961) give

$$\frac{c}{c_0} = \tfrac12\left[\operatorname{erfc}\frac{x-ut}{2\sqrt{Dt}} + e^{ux/D}\operatorname{erfc}\frac{x+ut}{2\sqrt{Dt}}\right].$$

After one hour the front is at 360 m and about 8.5 m wide.

What it checks

Advection without excess numerical diffusion. A first-order upwind scheme adds $u\,\Delta x/2$, already 2.5 × $D$ on a 0.5 m grid, and smears the front; a second-order TVD scheme should converge at order 2 on this smooth, monotone profile (minimum 1.6). The demo solver (MUSCL with van Leer limiter, central diffusion, SSP-RK2) reaches 1.67 across the four grids: its coarsest grid is half the front width, so the asymptotic order of 2 only shows on the finer pairs (1.8).

Citation

Ogata, A., and Banks, R. B. (1961). A Solution of the Differential Equation of Longitudinal Dispersion in Porous Media. USGS Professional Paper 411-A.

Sources

Ogata, A., Banks, R. B. (1961). A Solution of the Differential Equation of Longitudinal Dispersion in Porous Media. USGS Professional Paper 411-A.

Test A1P-030_advection_dispersion_tracer · version 1.0.0 · data licence analytic-public