(10) As requested by Roache, we use our approach to evaluate two new variants of the GCI method proposed by Oberkampf and Roy [18] (GCIOR ) and by Roache [1] (GCI3 ).
Display Formula

$UGCIOR={1.25|\u025b21rpth-1|,0.9\u2264P\u22641.13|\u025b21rmin(max(0.5,pRE),pth)-1|,0<P<0.9\u2003or\u2003P>1.1$

(2)

Display Formula$UGCI3={1.25|\epsilon 21rmin(pRE,pth)\u22121|,0.9\u2264P\u22641.13|\epsilon 21rmin(pRE,pth)\u22121|,0<P<0.9\u2009\u2009or\u2009\u2009P>1.1$

(3)

To address Roache’s concern of using

$pRE$ when

$pRE>>pth$, we also evaluate an alternative form of the FS method (FS1 method). The FS1 method is the same as the FS method for

$P<1$ but uses

$pth$ instead of

$pRE$ in the error estimate for

$P>1$. Thus, Eq. (14) in Ref. [

2] becomes

Display Formula$UFS1={[FS1P+FS0(1\u2212P)]|\epsilon 21rpRE\u22121|,0<P\u22641[FS1P+FS2(P\u22121)]|\epsilon 21rpth\u22121|,P>1$

(4)

Following the same procedure described in Sec. 2.4 of Ref. [

2],

$FS0=2.45$,

$FS1=1.6$, and

$FS2=6.9$ are recommended, and the final form of the FS1 method is

Display Formula$UFS1={(2.45\u22120.85P)|\epsilon 21rpRE\u22121|,0<P\u22641(8.5P\u22126.9)|\epsilon 21rpth\u22121|,P>1$

(5)

To compare the relative conservativeness between different verification methods, the three new methods are rewritten in terms of the same error estimate

$\delta RE$.

Display Formula$UGCIOR={1.25CF|\delta RE|,0.9\u2264P\u22641.13(rpRE\u22121)rmin(max(0.5,pRE),pth)\u22121|\delta RE|,0<P<0.9\u2009\u2009\u2009\u2009or\u2009\u2009\u2009P>1.1\u2009\u2009$

(6)

Display Formula$UGCI3={1.25(rpRE\u22121)rmin(pRE,pth)\u22121|\delta RE|,0.9\u2264P\u22641.13(rpRE\u22121)rmin(pRE,pth)\u22121|\delta RE|,0<P<0.9\u2009\u2009or\u2009\u2009\u2009P>1.1$

(7)

Display Formula$UFS1={(2.45\u22120.85P)|\delta RE|,0<P\u22641(8.5P\u22126.9)CF|\delta RE|,P>1$

(8)

The factors of safety for all the verification methods discussed so far are shown in Fig.

1. One problem of the GCI2 method is the jump of factor of safety across the asymptotic range at

$P=1$. For two grid-triplet studies with one at

$P=0.999$ and the other at

$P=1.001$, the factor of safety suddenly increases from 1.25 to 3 even though

$P$ only varies by less than 0.2%. Eça et al. [

19] gave similar comments on this issue: “However, it is not easy ‘to accept’ a jump of a factor of 2.4 in the uncertainty when the observed order of accuracy may vary by only 0.1.” Similar problems exist for the GCIOR and GCI3 methods when

$pRE$ differs from

$pth$ by 10%. It should be noted that the GCIOR method set the lower limit of

$pRE$ to be larger than 0.5, which corresponds to

$P\u22650.25$ for a nominal second order method. Thus, the factor of safety for

$P<0.25$ for the GCIOR method shown in Fig.

1 is only a result of the mathematical reformulation. Figure

1 also shows that the GCIOR and GCI3 methods are much more conservative than the other methods for

$0.25<P<0.9$ and coincide with the GCI2 method for

$P>1.1$. The FS1 method is less and more conservative than the FS method for

$1<P\u22641.235$ and

$P>1.235$, respectively.