A numerical solver returns a smooth curve. That appearance does not establish that it solved the intended equations, and solving the equations does not establish that they represent the target system. Separate verification of the computation from validation of the model.
Use x′=−x with x(0)=1 as a controlled reference. Its exact solution e^(−t) lets you measure numerical error directly, without fitting unknown physical parameters at the same time.
Predict
Forward Euler uses x next=(1−h)x. Predict what happens with h=1/2, h=1 and h=3. Which failures arise from the numerical step rather than from the exact decay model?
Experiment
At final time 1, compare two admissible step sizes with the exact reference e^(−1).
Step h
Number of steps
Euler value at 1
1/2
2
1/4
1/4
4
81/256
Exact
Not a time-step method
e^(−1)
The smaller step improves this example. Calculate absolute errors and their ratio. Do not expect an exact factor of two at coarse steps: asymptotic order describes a limiting regime with sufficient smoothness and stability, not every pair of resolutions.
Understand
Independent checks target different failure modes. An exact equilibrium tests whether a stationary state stays stationary. A conservation identity tests bookkeeping. An analytic transient tests dynamics. Step refinement tests numerical convergence, while positivity checks can reveal an overshoot even if an integral invariant remains correct. Tests that merely reproduce the implementation formula can share the same mistake and provide weak independent evidence.
If known errors satisfy E(h)≈Chᵖ, then log₂(E(h)/E(h/2)) estimates order p. When the exact answer is unavailable, differences between successive refinements can estimate behavior under additional asymptotic assumptions. Agreement of two implementations is not a proof if both use the same flawed algorithm or model.
Record absolute and relative tolerances with scales. A relative error is undefined or misleading near a zero reference value; combine it with an absolute tolerance. Distinguish discretization error, floating-point error, iterative-solver tolerance and Monte Carlo sampling error. Tightening one source does not automatically reduce the others.
Reproducibility needs the data version, code or complete pseudocode, environment, parameters, random generator and seed where used, stopping rules, tolerances and output-generation steps. A seed alone cannot reproduce a result after the code or input data change. Preserve failed runs that informed revisions, especially when they influenced the final method.
Bound the workload. A small investigation can specify a maximum grid size, finite parameter set and stopping condition, then report budget exhaustion as unresolved rather than as proof that no solution exists. The computational limit belongs in the evidence record.
Apply
Package one bounded computation with an explicit input table, reproducible steps and three independent checks. Ask another learner to reconstruct one reported result. Record any discrepancy and its cause. A successful rerun supports reproducibility of that computation; it does not independently validate the physical assumptions or establish research novelty.
Continue
The remaining lessons turn these modeling skills into a sustained mathematical investigation. Begin with a precise question and a literature map that preserves assumptions and separates known results from proposed extensions.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
Errors at step sizes h and h/2 are 1/8 and 1/32. What observed order log₂(Eh/Ehalf) results?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A numerical balance has initial 10, input 4, output 3 and final 10. What residual final−initial−input+output results?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Distinguish verification and validation for a numerical tank model.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Design three independent checks for an exchange-model solver.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
What should a reproducibility record contain beyond a random seed?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Package a bounded computation so another learner can rerun it.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
