What is solution verification?

Estimating how much error the numerical solution of a particular calculation (your model, grid and settings) contains. Unlike code verification, which checks the program is correct, it quantifies the reliability of the current result. The main errors are iterative error from truncating iterations, discretisation error from grid and time step, and round-off from finite precision.

How small must iterative error be?

Well below discretisation error — as a guide, at least an order of magnitude. In the example, with a residual tolerance of 10⁻³, iterative error grew in share as the grid was refined, and observed orders scattered as 2.77, 1.16 and −0.85; tightening to 10⁻⁶ brought them to 2.00. Tighten the tolerance when refining the grid.

When does round-off matter?

With very fine grids or single-precision arithmetic. In the example, double precision kept reducing error even at 20,000 cells, but single precision showed round-off from about 160 cells and error actually grew beyond 640. Run fine grids in double precision, and suspect round-off when refinement stops reducing error.

Solution Verification

Category: V&V – ASME Standards | Revised 2026-10-01

Theory: three errors in a numerical solution

Overview

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Refining the grid to get the convergence order, I got first order or even negative instead of the expected second. Is the scheme broken?

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Suspect iterative convergence before the scheme. Besides discretisation error from the grid, a numerical solution carries iterative error from stopping iterations early and round-off from finite precision. Refining reduces discretisation error, but with an unchanged tolerance iterative error doesn't shrink and may even grow. Once iterative error rivals discretisation error, observed orders scatter. Solution verification separates the three and estimates discretisation error alone.

Error decomposition

$$ \varepsilon_{total} = \varepsilon_{iter} + \varepsilon_{disc} + \varepsilon_{round}, \qquad \varepsilon_{iter} \ll \varepsilon_{disc} $$

Estimating discretisation error by grid comparison presupposes that iterative and round-off errors are small enough.

The example problem

$$ p_{obs} = \frac{\ln\left(e_{2h}/e_h\right)}{\ln 2}, \qquad -u'' = \pi^2\sin(\pi x),\ \ u = \sin(\pi x) $$

Solved with second-order central differences; observed order $p_{obs}$ follows from the error ratio when spacing is halved and should approach 2 if all is well.

Coffee Break Trivia

Separating verification from validation

Until the 1990s, discussions of computational credibility often confused whether a calculation solves its equations correctly (verification) with whether the equations represent reality (validation). In his 1998 book Roache separated the two clearly and further split verification into code verification, checking the program, and solution verification, estimating the error of individual calculations. This framework later became the backbone of the ASME V&V standards. Without first grasping numerical error through solution verification, a difference against experiment can't be attributed to the model or the numerics.

Worked examples

Example 1: effect of iterative tolerance

Using Gauss–Seidel, the stopping criterion (residual reduction relative to initial) was varied, and maximum error and observed order were found under refinement:

Tolerance	10 cells	20 cells (order)	40 cells (order)	80 cells (order)
10⁻³	7.3×10⁻³	1.1×10⁻³ (2.77)	4.8×10⁻⁴ (1.16)	8.7×10⁻⁴ (−0.85)
10⁻⁶	8.3×10⁻³	2.1×10⁻³ (2.01)	5.1×10⁻⁴ (2.00)	1.3×10⁻⁴ (2.01)
10⁻¹⁰	8.3×10⁻³	2.1×10⁻³ (2.01)	5.1×10⁻⁴ (2.00)	1.3×10⁻⁴ (2.00)

Converging the 80-cell case to 10⁻⁶ took about 9,000 iterations, and to 10⁻¹⁰ about 15,000. Finer grids need more iterations to reach the same tolerance.

Example 2: single-precision round-off

Cells	Double-precision error	Single-precision error
40	5.1×10⁻⁴	5.1×10⁻⁴
160	3.2×10⁻⁵	3.7×10⁻⁵
640	2.0×10⁻⁶	1.5×10⁻⁴
2,560	1.3×10⁻⁷	3.6×10⁻³
10,240	7.9×10⁻⁹	0.45

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In single precision, error grows as the grid gets finer!

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Refining multiplies subtractions of nearly equal values in the difference formulas, each losing significant digits. Single precision carries only about seven digits, so beyond a certain fineness round-off overtakes discretisation error. Double precision, with about sixteen digits, was fine to far finer grids here. Practical CFD and structural codes default to double, but beware when using single precision on GPUs for speed. If refining doesn't improve — or worsens — results, suspect round-off.