SCIENCE · INVESTIGATE AND EXPLAIN

Converge a numerical method without confusing it with validation

The numerical model assumes dv/dt=−v with initial10 m/s, evaluated at2 s. A separate synthetic observation comes from dv/dt=−lambdaTrue v. Keeping the exact assumed-model endpoint and independent observed endpoint distinct lets us separate numerical and physical-model discrepancies.

Learning outcome: Compare two grids against the exact assumed endpoint, then compare that exact endpoint with an independent changed-coefficient packet. Preserve signed error decomposition.

Starting recap

Differentiate/integrate with explicit initial conditions; identify each force recipient and the inertial frame. Preserve force dimensions, geometric constraints and the distinction between model and solver error.

All lessons are open directly. Your saved experiments remain when you leave and return.

Try

If lambdaTrue=1, the independent endpoint matches the exact assumed model and only discretization error remains. If lambdaTrue=.5, the observed endpoint is10e⁻¹≈3.67879 m/s, which a finer grid of the unchanged model will not approach.

Open the experiment

Predict

Convergence of a solver as its timestep shrinks establishes:

Your prediction and reason

Experiment

Compare two grids against the exact assumed endpoint, then compare that exact endpoint with an independent changed-coefficient packet. Preserve signed error decomposition.

Separate timestep convergence from a wrong force coefficient

Worked teaching model · synthetic values · Numerical model assumes dv/dt=−v with initial10 m/s; Heun integrates to2 s. Independent synthetic observation comes from dv/dt=−lambdaTrue v with the selected coefficient. Exact assumed-model and independent endpoints are kept separate; convergence is not model validation.

Heun steps over2 s: 4 steps Independent true decay coefficient: 1.25 s⁻¹

Reset keeps your prediction, captures and practice history.

Two independent comparisons at the same endpoint

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: Heun assumed coefficient1 (solid).2: exact assumed model (dashed).3: independent observation (dotted). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Two independent comparisons at the same endpoint

Heun assumed coefficient1

number of equal intervals	endpoint velocity (m/s)
2	2.5
4	1.525879
6	1.419139
8	1.387779
10	1.37448
12	1.367628
14	1.363641
16	1.361117
18	1.35942
20	1.358225

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

exact assumed model

number of equal intervals	endpoint velocity (m/s)
2	1.353353
20	1.353353

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

independent observation

number of equal intervals	endpoint velocity (m/s)
2	0.82085
20	0.82085

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Heun step size	0.5	s
Numerical assumed-model endpoint	1.525879	m/s
Exact assumed-model endpoint	1.353353	m/s
Independent observed endpoint	0.82085	m/s
Pure discretization error	0.1725261	m/s
Exact model minus independent observation	0.5325028	m/s
Total numerical minus observation	0.7050289	m/s

Refining a time grid can converge accurately to the wrong force model.

Heun averages the initial and predicted-end derivative; for dv/dt=−v the factor is1−h+h²/2. Compare with the exact assumed coefficient first to isolate numerical error, then with the independent coefficient packet to test the model. Total error is their signed sum; cancellation can make a coarse grid look deceptively better. A one-point discrepancy alone still does not uniquely identify a real force law.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Refining a time grid can converge accurately to the wrong force model.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Converge a numerical method without confusing it with validation

Heun’s method predicts an endpoint derivative, then averages it with the initial derivative for the step. For the assumed equation its one-step factor is1−h+h²/2. With N steps, h=2/N and vnum=10(1−h+h²/2)^N. The exact assumed-model result is10e⁻²≈1.35335 m/s.

At N4, the numerical result is1.5258789 m/s, a discretization error+.1725261. If the independent coefficient is1.25 s⁻¹, the observed endpoint is10e⁻²·⁵≈.8208500 m/s. The exact model itself differs from that observation by+.5325028; total numerical-minus-observed error is the signed sum of those two contributions.

Refining to N8 gives1.3877788 and N16 gives1.3611174 m/s, approaching the exact assumed model rather than the independent observation. A wrong coefficient does not become right because the solver converges. Signed error cancellation can also make a coarse result appear closer to an observation, so convergence and validation require different comparisons.

Changed case: predict before checking

Retained investigation

Compare two grids against the exact assumed endpoint, then compare that exact endpoint with an independent changed-coefficient packet. Preserve signed error decomposition. How could accidental cancellation make one coarse-grid result look deceptively credible?

Sources and boundaries

Primary conceptual reference checked30 October2026: OpenStax University Physics Volume1. Original mathematical cases and synthetic records; no performed experiments, practical certification or actual equipment safety claims. Numerical method/error relations are derived explicitly for the supplied equations. Preserve independent checks, conditional regimes and first-response evidence. No physical release, launch, loading or pulley procedure is requested.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

How could accidental cancellation make one coarse-grid result look deceptively credible?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Derive the unstable mode of an otherwise decaying equation Next: Integrate a force along two genuinely different paths

Save a science backup or import one

Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.

A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.

Choose a science backup (maximum 36 MB)

Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.

Complete file sets and course backups

A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.

Export scope

The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.

Choose the manifest and all parts together

0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.

Science storage: loading. No account or remote computation is required.