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advanced32 min studyLesson 930 in Physics

Verification, validation and discretization error

Estimate numerical convergence without turning it into a claim that a physical model is experimentally validated.

Before you begin
On this page

The question for today

What you’ll understand

  • Estimate observed order and a Richardson limit from three refinements.
  • Distinguish verification of equations from validation against physical observations.
  • Report deterministic error, statistical uncertainty and reproducibility data separately.

A credible numerical experiment answers two questions: does the calculation solve the specified mathematical problem adequately, and does that problem describe the physical observation of interest? Verification addresses the first; validation addresses the second within a stated application domain. Neither a small solver residual nor a smooth grid-refinement curve answers both. A report should identify the target observable, numerical approximation, physical assumptions and evidence needed for each claim, while keeping numerical error and statistical uncertainty distinguishable.

Verify complete equations and their implementation

Exact solutions provide useful benchmarks but may exercise only a simplified subset of a code's terms. Manufactured solutions instead begin with a chosen smooth field and derive the source, initial data and boundary conditions required to make it an exact solution. This can test forcing and boundary handling together. It cannot validate that manufactured forcing as a real experiment. An accidentally exact low-degree polynomial is valuable for detecting mistakes but may not reveal the expected asymptotic error order.

Consistency asks whether the discrete equations approach the intended differential equations as resolution improves. Stability asks how perturbations propagate. Convergence asks whether numerical solutions approach the intended solution in a specified norm and limit. Their relation depends on the problem and theorem assumptions, not a universal slogan. An elliptic algebraic solve also has an iteration error separate from spatial truncation, while time-dependent calculations must distinguish temporal and spatial refinement before inferring either order.

utt=c2uxx+f,u∗(x,t)=Asin⁡(πx/L)cos⁡(Ωt) ⟹ f=(c2π2L2−Ω2)u∗u_{tt}=c^2u_{xx}+f,\quad u_*(x,t)=A\sin(\pi x/L)\cos(\Omega t)\ \Longrightarrow\ f=\left(\frac{c^2\pi^2}{L^2}-\Omega^2\right)u_*
A is a displacement, L a length, c a speed and Ω an inverse time. Homogeneous Dirichlet conditions hold at x = 0,L; initial displacement is u*(x,0) and initial velocity is zero. f has displacement/time² units.↔ Scroll wide equations horizontally.

Infer order only in an asymptotic refinement regime

Suppose an observable has a leading error proportional to a power of one resolution parameter h. Three consistently refined calculations can estimate that power from successive differences, provided the leading term dominates and the differences are nonzero with appropriate signs. If time and space are refined together, the inferred order describes that refinement path. If boundary error, under-resolved structure, solver tolerance or roundoff dominates, the ratio may fail to approach the nominal scheme order.

Richardson extrapolation cancels the assumed leading term to estimate the limiting observable. It is not automatically a rigorous bound, and the limit remains a solution of the selected model and boundary conditions. A continuum-extrapolated pressure from an inaccurate material law can disagree with experiment for entirely physical reasons. Record the raw refinement values and the observed ratios so readers can assess whether a clean asymptotic trend, rather than a convenient fitted number, supports the reported estimate.

pobs=ln⁡ ⁣∣Qh−Qh/2Qh/2−Qh/4∣ln⁡2,Q∗≈Qh/4+Qh/4−Qh/22pobs−1p_{\rm obs}=\frac{\ln\!\left|\frac{Q_h-Q_{h/2}}{Q_{h/2}-Q_{h/4}}\right|}{\ln2},\qquad Q_*\approx Q_{h/4}+\frac{Q_{h/4}-Q_{h/2}}{2^{p_{\rm obs}}-1}
The formulas assume Qh = Q* + Ch^p plus smaller corrections along the stated refinement sequence, with nonzero leading differences. They do not independently certify that the asymptotic model applies.↔ Scroll wide equations horizontally.

Validate observations and preserve reproducibility

Validation compares the model with measurements under matched initial, boundary and operating conditions. Experimental uncertainty, uncertain material parameters, sampling variability and unresolved model discrepancy may all matter. A deterministic discretization bound is not automatically a Gaussian standard deviation and should not be combined in quadrature with a measured standard error without a justified probabilistic model. Agreement within an uncertainty interval supports a specified comparison; it does not prove a model universally correct.

Reproducibility requires the equation and units, geometry, mesh, step rules, tolerances, boundary and initial data, algorithm and software versions, precision, random loading or seeds when applicable, and the definition of each reported observable. Parallel reduction order can affect roundoff, especially in sensitive dynamics. The sampling lab shows exact independent-trial sample-mean distributions rather than fresh random runs. It illustrates a known statistical scaling, but cannot measure discretization error, rerun a fluid code or validate a physical experiment.

Var⁡(X‾n)=σ2nfor independent identically distributed samples,∣Qh−Q∗∣≤Bhis a distinct deterministic claim\operatorname{Var}(\overline X_n)=\frac{\sigma^2}{n}\quad\text{for independent identically distributed samples},\qquad |Q_h-Q_*|\le B_h\quad\text{is a distinct deterministic claim}
σ² is the population variance, while Bh is a declared deterministic bound if one has actually been established. An extrapolation estimate is not automatically such a bound, and correlated samples change the variance formula.↔ Scroll wide equations horizontally.

Work it through

Observed order from three refinements

A dimensionless observable has Qh = 1.04, Qh/2 = 1.01 and Qh/4 = 1.0025. Assume one leading power-law discretization term is justified along this refinement path. Find its observed order, Richardson limit and estimated finest-grid error.

  1. The successive differences are 0.03 and 0.0075. Their ratio is four, so the observed order is log(4)/log(2) = 2.

  2. The finest correction is (1.0025 − 1.01)/(4 − 1) = −0.0025. Adding it to the finest result gives an extrapolated limit of 1.

  3. The estimated finest error is therefore +0.0025 relative to that limit. This exact pattern is consistent with a quadratic leading term, but the three numbers alone do not rule out additional errors, verify boundary conditions or establish physical agreement with measurements.

The result

The observed order is 2, the extrapolated limit is 1 and the estimated finest-grid error is +0.0025 under the stated model.

The living notebook

Explore the model

Interactive lab

Interactive lab · sampling

Statistical scaling, separately from grid error

Reset and set BOTH Success probability p = 0.50 and Independent samples n = 25; predict sample-mean variance 0.01 and standard deviation 0.10. Reset and set p = 0.50 and n = 100; predict variance 0.0025 and standard deviation 0.05. These are exact binomial sample-mean distributions, not repeated random draws, grid-refinement results or physical validation data.

Statistical scaling, separately from grid errorExact binomial masses for the mean of n independent Bernoulli trials, compared with normal mass between adjacent half-count boundaries; normal tails outside the finite support remain outside the bars. Playback highlights outcomes, not random samples. The Berry–Esseen bound applies to standardized CDFs, not individual bars or continuity-corrected errors. At p = 0 or 1, normal comparison and that bound are disabled. Normal-CDF evaluation uses a numerical approximation with absolute error below 7.5×10⁻⁸ (at most 1.5×10⁻⁷ per interval, before rounding). Current readings: Mean sample proportion: 0.5; Sample-mean variance: 0.0125; Uniform CDF error bound: ≤ 0.1062. All parameters are adjustable with the labeled controls below.Exact distribution of the mean of 20 independent Bernoulli variables with success probability 0.5; playback highlights outcome 0.00.250.50.75100.0580.1150.1730.23Sample mean · k / nProbability massNormal bin masses are an approximationk = 0 · exact probability = 9.54e-7
Exact binomial massNormal interval massHighlighted outcome / mean
Mean sample proportion
0.5
Sample-mean variance
0.0125
Uniform CDF error bound
≤ 0.1062
t = 0 s

Exact binomial masses for the mean of n independent Bernoulli trials, compared with normal mass between adjacent half-count boundaries; normal tails outside the finite support remain outside the bars. Playback highlights outcomes, not random samples. The Berry–Esseen bound applies to standardized CDFs, not individual bars or continuity-corrected errors. At p = 0 or 1, normal comparison and that bound are disabled. Normal-CDF evaluation uses a numerical approximation with absolute error below 7.5×10⁻⁸ (at most 1.5×10⁻⁷ per interval, before rounding). Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA known sampling standard error and a deterministic numerical error require different evidence.

Pause & reason

Check your understanding

What conclusion can a verified asymptotic grid-refinement study support on its own?

Your turn

Try it yourself

An experiment reports a dimensionless mean 2.00 with Gaussian standard uncertainty 0.03. A computed value is 2.04, with an independently established deterministic numerical bound ±0.01 and no other uncertainty specified. Find the allowed continuum-model discrepancy range in units of the experimental standard uncertainty. Does this alone validate the model?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

First turn the numerical bound into the continuum interval 2.03 to 2.05. Treat that interval as a bound, rather than a new Gaussian variance.

Compare with the solution

The continuum-model value lies between 2.03 and 2.05 under the declared numerical bound. Its difference from the reported experimental mean therefore lies between 0.03 and 0.05, or between 1 and about 1.67 experimental standard uncertainties. The bound was not a random standard deviation, so adding 0.01² to 0.03² would require an extra probabilistic assumption. This comparison alone neither validates nor rejects the model: the intended use, experimental conditions, other parameter uncertainties and a prespecified comparison criterion still matter.

Keep the idea

Take it with you

  • Verification and validation answer different questions.
  • Observed order and extrapolation need an asymptotic error model.
  • Deterministic bounds, statistical uncertainty and model discrepancy should be reported distinctly.

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