Objetivación de la honradez
5.2. Licitaciones transparentes
Error estimation techniques used in the FV framework can be divided into four main groups:
1. Truncation analysis. Every smooth solution can be written in a form of a
Taylor series expansion about a given point in space. The series is potentially infinite depending on the number of non-zero gradients, thus:
Φ(x) = φP + ∞ X n=1 1 n!(x−xP) n :: |{z} n (∇∇| {z }..∇ n φ)P. (4.2)
The FVM approximates the solution by using a truncated form of the Taylor series. The error in the solution is then equal to the sum of truncated terms:
e(x) = Φ(x)−φ(x) = ∞ X n=p 1 n!(x−xP) n :: |{z} n (∇∇| {z }..∇ n φ)P, (4.3)
wherep is the order of discretisation. The error can be estimated by approxi- mating higher gradients from the existing FV solution and inserting them into Eqn. (4.3).
One of the first error estimation techniques used in the framework of the FVM is Richardson Extrapolation (RE). Examples can be found in [23, 46, 68, 101] etc. The RE method requires the existence of solutions of the problem on two or more meshes with different cell sizes to estimate the leading term of the truncation error by solving the system:
Φ(x)−φ1(x) =C hp1,
Φ(x)−φ2(x) =C hp2, (4.4) where C hpi is the leading term of the truncation error, Φ(x) is the unknown exact solution and φi(x) is the FV solution on a mesh with density hi. hi is
defined as a ratio between the volume and area of the cell [68]: hi =
V
P
f|S|
4.2 Literature Survey 79 By solving the system in Eqn. (4.4),C and Φ(x) can be obtained and the error can be expressed as follows:
e=|Φ(x)−φ2(x)|= |φ2(³x)−φ1(x)| h1 h2 ´p −1 . (4.6)
This is not a reliable method because it generally under-estimates the error, especially if the finer mesh is still coarse [68, 95]. Another drawback is the fact that it requires two solutions which make it time consuming.
Methods which can estimate the error by using a single solution obtained on a single mesh started appearing relatively recently [10, 39, 55, 63, 68, 70, 138, 159], etc.. A method for the first-order accurate FVM discretisation presented by Ilinca et al. in [63] estimates the error as the difference between the computed solution, assumed to be of constant value within the CV, and a reconstructed linearly-varying solution, thus:
e= sZ VP [(φP + (x−xP)·(∇φ)P)−φP]2dV = sZ VP (x−xP)2dV : (∇φ)2P = q M : (∇φ)2 P, (4.7)
where M is the moment of inertia of the CV and (∇φ)P is the reconstructed
gradient at the node P. The estimator accounts for the flow physics through the gradient of φ and it also accounts for the size and shape of the control volumes through the moments of inertia. The same approach was followed by Jasak in [68, 70] where the error is estimated using the second-order term in the Taylor Series and applied to second-order accurate FVM:
e= 1 2 Z VP [(x−xP)2 : (∇∇φ)P]dV = 1 2M: (∇∇φ)P, (4.8)
where (∇∇φ)P is calculated from the available solution. This estimator tends
to under-estimate the error when the computational mesh is coarse because the higher-order terms neglected by the estimator are still of considerable importance.
An interesting method for error estimation based on truncation analysis can be found in [10, 39, 55, 138]. The error is computed on mesh edges as follows:
e=
Z l
0
q
s(l)TH¯(l)s(l)dl (4.9)
where s(l) is parametric representation of the edge and ¯H(l) is a modified Hessian obtained from the second gradient. The estimator is used to drive a mesh adaptive algorithm aimed at equidistributing the error over every edge in the domain.
An approach to error estimation where the convection and diffusion fluxes are calculated using a fourth-order scheme and compared with the ones from the second-order accurate FV solution was presented by Muzaferija et al. in [101, 102]. The source of the truncation error τP is defined as:
τP =
X
f
[(ρU·S)f(φhf −φf)−(ρΓ)f((S· ∇φ)hf −(S· ∇φ)f)] (4.10)
where φh
f and (S· ∇φ)hf are evaluated from the FV solution using a fourth-
order scheme while φf and (S· ∇φ)f are known from the FV discretisation
process. The authors assumed that τP serves as a source which can be added
to the discretised form of the governing differential equation in order to obtain the exact solution. This method under-estimates the error and is even less accurate than Richardson Extrapolation because it cannot detect non-local phenomena. A similar approach to error estimation developed for hyperbolic equations, for which the error may manifest itself away from its source, was presented by Zhang et al. in [159]. The source term of the error equation consists of the higher order truncated terms which are evaluated from the existing solution. The estimated errors are in good agreement with the exact errors in terms of their locations but the magnitudes are under-estimated. It has also been shown that an efficient grid adaptation can be achieved using the estimated error source to drive the adaptive algorithm instead of the error distribution itself, especially on cases where the error is convected away from its source.
2. Methods based on derived quantities of the solution variable. The
second group of estimators used in the FVM framework was derived upon a fact that the discretisation schemes enforce conservation of the primary quantities
4.2 Literature Survey 81 for which the equations are being solved, regardless of the mesh resolution, but none of them conserve their derived properties (i.e.kinetic energy (1
2U·U) and angular momentum (x×U) are derived from velocity) except when the mesh is sufficiently fine. Haworth et al. [57] presented a method which measures cell-to-cell imbalances of the angular momentum and kinetic energy in order to characterise the solution error. The imbalance of kinetic energy can be written: Res(1 2U·U) =∇ ·(ρ( 1 2U·U)U) +∇ ·(P U)−P∇ ·U +∇ · µ 2 3µ(∇ ·U)U ¶ − ∇ ·£µ¡∇U+ (∇U)T¢ ·U¤ +∇U: · µ¡∇U+ (∇U)T¢+ µ 2 3µ∇ ·U ¶ I ¸ , (4.11) and the imbalance of angular momentum can be written as follows:
Res(x×U) =∇ ·[ρ(x×U)U]−x×(∇ ·σ)−ρ(x×g), (4.12) where all fields are known from the FV solution and the terms are evaluated using the same approximation as the ones used for the FV solution. The drawback of the method is that it is not capable of estimating the absolute error levels in the velocity field. Jasak [68] generalised the same approach and made it capable of estimating absolute error levels. The error is estimated by calculating the imbalance in the transport equation for the second moment of φ defined asmφ= 12φ2. The equation for the second moment can be obtained
by multiplying the general transport equation Eqn. (2.32) with φ, thus [68]: Res(mφ) =
Z
ΩP
[∇ ·(ρUmφ)− ∇ ·(ρΓφ∇mφ)−Sφ(φ)φ+ρΓφ(∇φ.∇φ)]dV.
(4.13) The values of all fields in the above equation are known and the imbalance is calculated using the same approximation as the ones used for FV equations. Multiplying Eqn. (4.13) with a normalisation factor enables recovery of the error magnitude, thus:
e= 2
s
|Res(mφ)|Ttrans
VP
. (4.14)
VP is the cell volume and Ttrans is the characteristic time scale of mφ defined
as:
Ttrans =
h2
|U|h+ Γφ
whereh is a cell size defined in Eqn. (4.5), |U| is a transport cell velocity and Γ is the diffusion factor. A problem with this error estimator is its inaccuracy on fine meshes. This is due to the higher order of variation of the second moment. The transport equation for the higher moment has to be discretised in a higher-order manner than the transport equation for the property to scale the errors correctly. It may happen the error estimate indicates the error exists even if it is zero. This undesired behaviour is the most severe in the regions where the variation of the variable is modest while in the regions of steep gradients it does not have a significant influence on the estimated error.
3. Residual methods for error estimation in the FV analysis have their origin
in similar work in the Finite Element field. In the FVM the concept of residual error estimation has been introduced by Jasak [68] and Angermann [12, 13]. The residual is a function measuring how well are the governing differential equations approximated over the computational cell and is defined as [68]:
Res(φ) = SuVP +Sp φPVP −X f Ff(φP + (xf −xP)·(∇φ)P) + X f (ρΓ)f(Sf ·(∇φ)P), (4.16)
where all fields are known from the available FV solution. The residual is consistent with the discretisation practice and is zero when the solution varia- tion is linear. It does not give information about the error magnitude directly because of its different dimension from φ. Jasak [68] has proposed a normali- sation practice to extract the magnitude of the error from the residual which consists of the convection and diffusion transport coefficients and the linearised part of the source term Sp, thus:
Fnorm = X f max(F,0) +X f · |S|(ρΓφ)f |d| ¸ +SpVP. (4.17)
and the error is calculated from:
e= Res(φ) Fnorm
(4.18) The normalisation practice assumes first-order accurate UD discretisation of the convection term, the first term on the r.h.s. in Eqn. (4.17), which en- sures that the convection contribution to the normalisation factor is always positive. This has an adverse effect on the accuracy of error estimation espe- cially on meshes with insufficient resolution where it may result in an incorrect
4.3 Error Transport Through a Face 83