Verification and Validation

Distinguishes numerical and physical errors through verification, validation, and convergence.

Conceptual context

A simulation may produce precise numbers and still be wrong. Verification asks whether the equations were solved correctly; validation asks whether those equations represent the physical system adequately. Both require references, refinement studies, and careful error interpretation. $y_sim$ must be compared with $y_ref$, but their difference may include numerical error, experimental uncertainty, and model inadequacy.

🟢 Essential level

Verification checks whether the numerical method behaves as intended. If $E_h$ decreases when $h$ is reduced, the algorithm shows convergence. Validation compares $y_sim$ with $y_ref$ to judge whether the model represents the phenomenon adequately. Error may be measured in original units or relative to the reference scale. The theoretical order states how it should fall, while the observed order describes measured reduction. A solution may converge and still be physically wrong when equations, parameters, or boundary conditions are inadequate. Verification therefore concerns computational correctness, whereas validation concerns physical credibility.

The two questions must remain separate. Stable computation describes algorithmic behavior, while experimental agreement describes model behavior. Both forms of evidence are necessary, yet neither can replace the other. This distinction prevents polished numerical output from being mistaken for physical truth.

A well-planned simulation project documents both forms of evidence separately: it first shows that the code converges toward the solution of its own equations, and only afterward compares that solution with independent data to judge its physical relevance.

🔵 Formal level

Basic formulas

Direct comparison between simulation and reference is written through:

Absolute error

$E_a$ retains the physical dimension of the result and is nonnegative. To compare discrepancies across scales, use:

Relative error

$E_r$ is dimensionless and becomes unreliable when $y_ref$ is zero or very small.

Refinement-based verification follows:

Convergence law

Here $E_h$ depends on $h$, $C$, and $p$. In the asymptotic regime, reducing $h$ produces a predictable error decrease. On a log-log graph, the slope approaches the convergence order.

Real student questions

Does a finer mesh always improve the result? No. Roundoff, instability, or an inadequate reference may dominate.

Does convergence prove the model is correct? No. It shows only that the numerical solution stabilizes for the chosen model.

Which error should I use? $E_a$ retains units; $E_r$ compares scales except near a zero reference.

How many grids are needed? At least three levels help determine whether the estimated order stabilizes.

Does a good fit validate the model? Only within the tested domain, uncertainty, and conditions.

Final synthesis

Verification demonstrates that computation solves the model correctly. Validation demonstrates that the model represents reality adequately. Refinement, references, and uncertainty turn computed numbers into scientific evidence.