FUNDACIÓN NACIONAL PARA EL DESARROLLO

a ₋ posteriori error estimates for convection-dominated problems. In [13] Angermannet al. the method was modified for density-driven time dependent problems. The proposed estimate is assembled from four parts which repre- sent the local indicators. The full error estimate is defined as a sum of local error indicators. The first indicator is the residual and the other indicators account for the imbalance of convection fluxes over the computational cell. This method is not easy to understand and it does not measure the absolute value of the error. It is also not accurate enough for engineering application because of severe over-estimation of the error.

4. The Equilibrated residual method is a technique which was used first

in the framework of FEM. Its extension to FVM was presented by Jasak in [68, 74]. The method is not as successful as its FEM counterpart because the procedure used for solving local problems is not accurate enough, especially on coarse meshes.

Even though the FV error estimation has undergone a rapid development during the last decade there is still some room for improvement. Truncation methods which can estimate the error from a single solution have been developed, but they remain prone to under-estimation of the error. Residual based methods are considered to be the most promising group of methods and will be utilised in this study. In order to be accurate, residual methods require conserved error fluxes through the cell faces and a reliable procedure for extracting the error magnitude from the residual. The approach taken during this study is aimed towards developing an estimator on mesh faces where balanced error fluxes and the data needed for extraction of error magnitude are available.

4.3

Error Transport Through a Face

The FV discretisation practice is conservative for each CV and is performed by approximating fluxes through the CV faces such that they are in balance with the sources/sinks acting within the CV. The approximated fluxes may contain errors which are transported through cell faces and these errors influence the accuracy in the neighbouring cells.

Due to the property of the FV discretisation, the general transport equation, Eqn. (2.32), can have three equally valid solutions on every internal face: two ex- trapolated from both sides of the face (φf)P, (φf)N, Fig. 4.1, and the interpolated

value onto the face φf [68].

P

N

φ

φ

( )φ

φ

(

)

φ

f

Figure 4.1: Inconsistency of the interpolated values on the face

The value on the face extrapolated from the cell P can be calculated using a second-order accurate variation of φ in the cell, Eqn. (3.3), such as [68]:

(φf)P =φP + (xf −xP)·(∇φ)P, (4.19)

while the value from the cell N can be obtained in the same manner as for the cell P, thus:

(φf)N =φN + (xf −xN)·(∇φ)N. (4.20)

The interpolated value on the face can be calculated using Eqn. (3.12) and the gradients in the cells P and N can be evaluated using the second-order accurate methods, Eqs. (3.24) and (3.28). When the variation of the solution is linear, the gradients in the cells and on the face are equal and the extrapolated values (φf)P

and (φf)N become equal to the interpolated value φf resulting in the discretisation

error equal to zero [68].

A transport equation for error can be derived by substituting Eqn. (1.2) into the transport equation Eqn. (2.32). For steady state conditions it reads:

Z VP ∇ ·(ρU(E(x) +φ(x)))dV ₋ Z VP ∇ ·(ρΓ_∇(E(x) +φ(x)))dV = Z VP Sφ(φ)dV. (4.21)

4.3 Error Transport Through a Face 85 The transient term is omitted as the focus here is on the discretisation error arising from the spatial terms. The terms can be reordered such that the terms known from the FV solution are put on the right-hand side and denoted by Res(φ), and the terms governing the error transport are put on the left-hand side. The transport equation for error then has the form [6]:

Z VP ∇ ·(ρUE(x))dV ₋ Z VP ∇ ·(ρΓ(_∇E(x)))dV =Res(φ) (4.22) where the residual term Res(φ) on the right-hand side has the following form:

Res(φ) =

Z

VP

[Sφ(φ)− ∇ ·(ρUφ(x)) +∇ ·(ρΓ(∇φ(x)))]dV. (4.23)

The residual acts as a source for the error, Eqn. (4.22), and has the same form as the transport equation Eqn. (2.32). Its approximated form reads [68]:

Res(φ) =Sφ(φ)VP − X f Ffφf + X f (ρΓ)f(S· ∇φ)f. (4.24)

If the evaluation of φf and (S· ∇φ)f terms in the residual is performed using same

procedures as for the convection Section 3.3.1 and the diffusion term Section 3.3.2, respectively, the residual is zero even though the error exists. This is a consequence of conservation which is then satisfied. But φf can also be approximated by us-

ing Eqn. (4.19) which is valid within the computational cell and (S_{· ∇}φ)f can be

approximated by:

(S_{· ∇}φ)f = (Sf ·(∇φ)P). (4.25)

When the variation of the solution is of higher-order than linear the residual evalu- ated by using Eqn. (4.19) and Eqn. (4.25) [68]:

ResP(φ) =Sφ(φ)VP− X f Ff(φP+(xf−xP)·(∇φ)P)+ X f (ρΓ)f(Sf·(∇φ)P), (4.26)

may not be equal to zero if the mesh is not sufficiently fine. Thus, it provides an estimate of the magnitude of the error source in the computational cell. Regarding the source term, the discretised source term Section 3.3 cannot be written as sum over cell faces and its influence on the discretisation error is neglected. The residual is consistent with the discretisation practice and it is zero when the variation of φ is linear. It is sensitive to the shape of the cells and should therefore capture the mesh induced errors because the extrapolated values onto the faces are dependent on (xf −xP).

As was shown by Ainsworth et al. [3, 8] and Jasak [68] the residual can be split into a sum of partial face residuals from all the faces bounding the computational cell, thus: ResP(φ) = X f ResP f(φ), (4.27)

where the partial face residualResP

f(φ) is defined as [68]:

ResPf(φ) =Ff(φf−(φP+(xf−xP)·(∇φ)P))+(ρΓ)f((S·(∇φ)P)−(S· ∇φ)f). (4.28)

Here,φf represents the interpolated value on the face shown in Fig. 4.1 and (S· ∇φ)f

is the surface normal gradient on the face calculated using Eqn. (3.48). The partial face residual represents the imbalance in values and gradients on the face originating from cell P Fig. 4.1, only. The partial face residual can detect how well is the governing equation satisfied on the faces comprising the cell and can therefore give information about the direction in which the error is transported.

When the left-hand side of Eqn. (4.22) is discretised in a second-order manner for a representative cell, we get:

X f FfePf − X f (ρΓ)f(Sf ·(∇e)Pf) = X f Ff(φf −(φP + (xf −xP)·(∇φ)P)) + X f (ρΓ)f((S·(∇φ)P)−(S· ∇φ)f) (4.29) from which it follows that:

eP

f =φf −(φP + (xf −xP)·(∇φ)P),

(S_{· ∇}e)P

f = (S· ∇φ)f −(S·(∇φ)P). (4.30)

After some algebra a link between the error and the error in gradient on the face can be found, thus:

eP

f = (xf −xP)·(∇e)Pf. (4.31)

From Eqn. (4.30) an interesting property of the error field can be found. The error can be transported in the opposite direction to the solution fluxes, because from Eqn. (4.30) it is clear that the convection FfePf and diffusion (Sf · (∇e)Pf) error

fluxes may have different signs from the fluxes Ffφf and (ρΓ)f(S· ∇φ)f used to

approximate convection and diffusion terms. The following section will show how this can be used to develop an error estimate.