VVUQ · Module 3 of 10
Solution Verification: Richardson Extrapolation and the GCI
A verified code still gives an approximate answer on any finite mesh. Solution verification estimates that discretization error, extrapolates to estimate the continuum-limit value, and reports a discretization-uncertainty band as a grid convergence index with a safety factor.
Readiness check
Learning objectives
- Calculate a Richardson estimate of the continuum-limit value from a grid triplet.
- Calculate the grid convergence index and interpret it as a conditional discretization-error estimate under the stated asymptotic assumptions.
- Explain the role of the GCI factor of safety without treating the GCI as an automatic bound.
- Evaluate whether the grids lie in the asymptotic range where these estimates are valid.
This module quantifies discretization error. Tick only what you can do closed-notes.
- Recall the observed order of accuracy from two grids.
- Recall Richardson extrapolation from numerical methods.
- Compute a relative difference between two results.
- Recall that a finer mesh has less error.
- Apply a factor of safety to an estimate.
The core idea
Solution verification estimates the discretization error of a specific result. Richardson extrapolation uses results from systematically refined grids to estimate the continuum-limit value. The GCI uses the observed grid convergence and a safety factor to report a spatial-discretization-uncertainty estimate for the selected grid solution.
fexact ≈ ffine + (ffine − fcoarse)/(rp − 1)GCI = Fs |ε| / (rp − 1)ε = relative difference between gridsEven a verified code produces only an approximate answer on any finite mesh; the gap to the exact solution is the discretization error, and estimating it is solution verification. Richardson extrapolation is the key tool: if the error falls at order p, then two solutions on grids differing by a refinement ratio r can be combined to estimate the continuum-limit value, fexact ≈ ffine + (ffine − fcoarse)/(rp − 1). With two grids and an assumed order, this is a conditional estimate, and the GCI applies a safety factor to report a spatial-discretization-uncertainty estimate for the fine-grid result. In the asymptotic range the extrapolated value is typically more accurate than either grid alone; outside it, the extrapolation can be unreliable. Keep three things separate: the extrapolation gives a conditional estimate of the continuum-limit value under an assumed convergence model; the difference between the fine-grid result and that estimate is an estimate of fine-grid discretization error; and the GCI applies a safety factor to report a discretization-uncertainty band for the grid solution. The GCI is not, by itself, an uncertainty interval for the extrapolated value. To report the error conservatively rather than optimistically, Roache's grid convergence index (GCI) wraps this estimate in a factor of safety: GCI = Fs|ε|/(rp − 1), where ε is the relative difference between the two grids and Fs is typically 1.25 for three or more grids. The GCI is a percentage numerical uncertainty band on the fine-grid result, the quantity that feeds the validation uncertainty later. All of this holds only in the asymptotic range, where the observed order is stable, so a convergence study uses systematically refined grids and checks that the order is consistent.
The skills, taught in order
Five skills build discretization-error estimation, extrapolation, and the GCI.
This module focuses on spatial discretization error and uncertainty estimated through systematic mesh refinement. A complete solution-verification assessment may also need to consider iterative, roundoff, and temporal discretization effects.
3.1 Discretization error
Discretization error is the difference between the finite-mesh result and the exact solution of the equations. Written with an explicit sign convention, the estimated signed discretization error of a grid solution is eh,est = fh − f0,est, so a negative value means the grid result lies below the estimated continuum-limit value. It is a numerical error, part of verification, and must be estimated before a result is compared to data, so validation is not corrupted by an uncorverged mesh.
3.2 Richardson extrapolation
With the observed order p and a refinement ratio r, Richardson extrapolation estimates the continuum-limit value from two grids: fexact ≈ ffine + (ffine − fcoarse)/(rp − 1). The correction term is the negative of the estimated signed discretization error of the fine grid: adding it moves the fine-grid result onto the extrapolated estimate, so its magnitude is the estimated error and its sign is opposite.
3.3 The grid convergence index
The GCI converts the extrapolation into a conservative uncertainty: GCI = Fs|ε|/(rp − 1), with ε the relative difference between grids and Fs a factor of safety (1.25 with three or more grids, 3 with two). It is reported as a percentage band on the fine result.
| Quantity | Expression | Meaning |
|---|---|---|
| Relative difference | ε = (ffine − fcoarse)/ffine | grid-to-grid change |
| Extrapolated value | ffine + (ffine − fcoarse)/(rp − 1) | estimate of the continuum limit |
| GCI | Fs|ε|/(rp − 1) | numerical uncertainty band |
The grid convergence index reports discretization error conservatively, as a percentage band on the fine-grid solution.
3.4 The asymptotic range
Richardson extrapolation and the GCI are valid only in the asymptotic range, where refining the grid gives a stable observed order matching the theoretical one. A study with at least three systematically refined grids allows the observed order to be estimated and provides a check of whether the results are consistent with asymptotic convergence.
3.5 Reporting numerical uncertainty
The output of solution verification is the fine-grid result plus or minus its GCI, a numerical uncertainty that becomes one input to the validation uncertainty. A mesh-dependent result reported without a discretization-uncertainty estimate does not quantify its sensitivity to spatial discretization. The GCI is one established method for reporting such an estimate. Keep the terms distinct: an error is a signed difference from an exact or estimated reference, while an uncertainty is a plus-or-minus interval intended to bound the unknown value or error.
Engineering connection: a CFD drag coefficient, structural displacement, reaction force, strain energy, or a stress measure demonstrated to be nonsingular and convergent is not considered complete until a grid convergence study gives it a numerical uncertainty band. Before applying Richardson extrapolation or the GCI, confirm that the selected quantity of interest is expected to approach a finite asymptotic value. A singular peak stress, such as one at a re-entrant corner, is not suitable for this treatment.
Worked example 1: Richardson extrapolation
A quantity is 0.940 on a coarse grid and 0.970 on a grid refined by r = 2, with an assumed order p = 2. Estimate the continuum-limit value by Richardson extrapolation.
- ProblemEstimate the continuum-limit value for the grids in Figure 1, using the assumed order p = 2.
- Given / findfcoarse = 0.940, ffine = 0.970, r = 2, p = 2. Find fexact.
- AssumptionsTwo grids with an assumed order p = 2, taken from the scheme rather than observed. Two solutions cannot demonstrate that the grids lie in the asymptotic range, so the extrapolation is conditional on the leading-order convergence model holding here. A study with at least three systematically refined grids would allow the order to be estimated and would give stronger evidence.
- Modelfexact ≈ ffine + (ffine − fcoarse)/(rp − 1).
- Equationsrp − 1 = 22 − 1 = 3correction = (0.970 − 0.940)/3
- SolveCorrection = 0.030/3 = 0.010. fexact ≈ 0.970 + 0.010 = 0.980.
- CheckThe correction (+0.010) has the magnitude of the estimated fine-grid discretization error and the opposite sign, since eh,est = fh − f0,est = −0.010. It is one-third of the 0.030 grid-to-grid change, as rp − 1 = 3 requires.
- ConclusionThe estimated continuum-limit value is f0,est = 0.980. Using eh,est = fh − f0,est, here eh,est = −0.010, indicating that the fine-grid result is 0.010 below the estimated continuum-limit value. Under the asymptotic assumption, extrapolation gives a sharper estimate than either grid alone; it does not make that estimate exact.
Worked example 2: the grid convergence index
For the same two grids (fcoarse = 0.940, ffine = 0.970, r = 2, p = 2), report the fine-grid result's numerical uncertainty as a GCI. Two grids means the two-grid factor of safety, Fs = 3.
- ProblemReport the GCI numerical uncertainty for the fine grid in Figure 2.
- Given / findfcoarse = 0.940, ffine = 0.970, r = 2, assumed p = 2, Fs = 3. Find the GCI.
- AssumptionsOnly two grids, so the order p = 2 is the scheme's theoretical order, assumed rather than observed. Two solutions cannot demonstrate that the grids are in the asymptotic range, which is why the two-grid factor of safety Fs = 3 applies rather than the 1.25 used when three grids confirm the observed order.
- Modelε = (ffine − fcoarse)/ffine; GCI = Fs|ε|/(rp − 1).
- Equationsε = (0.970 − 0.940)/0.970 = 0.0309GCI = 3 × 0.0309 / 3
- SolveGCI = 3 × 0.0309/3 = 0.0309 = 3.09%. The fine result is 0.970 ± 3.09%.
- CheckThe GCI (3.09%) exceeds the bare extrapolated error (about 1.03% of 0.970) by the factor of 3, giving a conservative band as intended. Adding a third grid to confirm the observed order would justify the smaller 1.25 factor and a band near 1.29%: the extra grid buys a tighter honest number.
- ConclusionThe fine-grid result carries a numerical uncertainty of about 3.1%. This band, not the bare number, is what a validation and decision downstream must use.
Misconceptions and diagnostics
| Mistake | Symptom | Diagnostic question | Correction |
|---|---|---|---|
| Extrapolating outside the asymptotic range | Unstable observed order | "Is the order consistent between grids?" | Refine until the order stabilises. |
| No factor of safety | Optimistic error estimate | "Did I apply Fs?" | Use the GCI with its safety factor. |
| Reporting a bare fine result | No numerical uncertainty | "Where is the GCI band?" | Report the result plus or minus its GCI. |
| Confusing GCI with validation error | Numerical band called model error | "Is this numerical or physical?" | GCI is numerical; validation compares to data. |
Practice ladder
For fcoarse = 1.00, ffine = 1.06, r = 2, p = 2, find the Richardson-extrapolated value.
Show answer
Correction = (1.06 − 1.00)/(4 − 1) = 0.06/3 = 0.02. fexact ≈ 1.06 + 0.02 = 1.08.
For those grids, find the GCI. Two grids, so use Fs = 3.
Show answer
ε = 0.06/1.06 = 0.0566. GCI = 3 × 0.0566/3 = 0.0566 = 5.66%.
Two grids give 2.10 and 2.00 with r = 2 but an observed order p = 1. Find the extrapolated value.
Show answer
rp − 1 = 2 − 1 = 1, so correction = (2.10 − 2.00)/1 = 0.10, fexact ≈ 2.20. A first-order scheme extrapolates with a larger correction.
A colleague reports a single-mesh CFD result with no uncertainty. Explain what a grid convergence study would add and what to report.
What good work looks like
Run three systematically refined meshes, check that the observed order is consistent with asymptotic convergence, extrapolate to estimate the continuum-limit value, and report the fine result with its GCI band, so the number carries a defensible numerical uncertainty.
Working with AI, and proving it yourself
Use AI as an examiner, not a solver
Portfolio task
Run a three-grid convergence study on a real simulation, extrapolate to estimate the continuum-limit value, and report the fine result with its GCI numerical uncertainty.
Retrieval and spaced review
Closed notes. Answer out loud, then reveal.
1. What is discretization error?
The difference between the finite-mesh result and the exact solution of the equations.
2. Write the Richardson estimate of the continuum-limit value.
fexact ≈ ffine + (ffine − fcoarse)/(rp − 1).
3. Write the GCI.
GCI = Fs|ε|/(rp − 1), a conservative numerical uncertainty.
4. Why a factor of safety?
To report the discretization error conservatively rather than optimistically.
5. When are these valid?
Only in the asymptotic range, where the observed order is stable.