ESTIMATE OF THE TRUNCATION ERROR OF FINITE VOLUME DISCRETISATION OF THE NAVIER-STOKES EQUATIONS ON COLOCATED GRIDS.

Transcription

1 ESTIMATE OF THE TRUNCATION ERROR OF FINITE VOLUME DISCRETISATION OF THE NAVIER-STOKES EQUATIONS ON COLOCATED GRIDS. Alexandros Syrakos, Apostolos Goulas Departent of Mechanical Engineering, Aristotle University of Thessaloniki, Greece. Abstract A ethodology is proposed for the calculation of the truncation error of finite volue discretisations of the incopressible Navier Stokes equations on colocated grids. The truncation error is estiated by restricting the solution obtained on a given grid to a coarser grid and calculating the iage of the discrete Navier Stokes operator of the coarse grid on the restricted velocity and pressure field. The proposed ethodology is not a new concept but its application to colocated finite volue discretisations of the incopressible Navier Stokes equations is ade possible by the introduction of a variant of the oentu interpolation technique for ass fluxes where the pressure-part of the ass fluxes is not dependent on the coefficients of the linearised oentu equations. The theory presented is supported by a nuber of nuerical experients. The ethodology is developed for two-diensional flows, but extension to three-diensional cases should not pose probles. Key words: finite volue, truncation error, colocated grids, oentu interpolation. 1. Introduction Finite volue ethods, and especially those of 2 nd -order accuracy, are very popular for the solution of the Navier-Stokes equations because, by today s standards, they offer acceptable accuracy on reasonably dense grids while being easy to ipleent. The truncation error is the easure of the discrepancy between the discrete syste which arises fro application of the finite volue ethodology and the original integral-differential equations. For structured grids several truncation error estiators have been proposed for particular discretisation schees, for exaple in [1], [2] and [3]. They express the leading ter of the truncation error in ters of the derivatives of the flow variables and the geoetry of the grid. These estiators are useful but they have the disadvantage that they are different for each discretisation schee, and that they apply in the case of structured grids which have been constructed fro distributions of diensionless variables with continuous derivatives. For ore general cases, a nuber of truncation error indicators have been proposed, as in [4], [5] or [6], that is, quantities which reseble the truncation error and soeties have the sae units, and are likely to be large in regions where the truncation error is large. However, these quantities ay fail to capture certain parts of the truncation error, for exaple the skewness-induced part if the indicator is constructed fro a one-diensional analysis as in [4]. Besides, an estiate would be ore useful rather than an indication. Multigrid solution ethods which use the full approxiation schee (FAS) autoatically provide a quantity, the relative truncation error between the finest and the iediately coarser grid, which can easily be converted into a truncation error estiate if the order of the discretisation is known see [7], [8], [9]. In fact, the truncation error estiator can be used independently of the ultigrid procedure. All that is required is a solution on a given grid, and a coarser grid which is siilar to the fine grid. The equations need not be solved on the coarse grid. This inspired the present authors to ipleent this estiator in the case of the finite volue discretisation of the incopressible Navier-Stokes equations on colocated grids. This is the peer reviewed version of the article published in final for in the International Journal for Nuerical Methods in Fluids, vol. 50, p (2006) DOI: /fld This article ay be used for non-coercial purposes in accordance with Wiley Ters and Conditions for Self-Archiving.

2 2. Finite volue discretisation: Basic principles and notation Here it will be useful to introduce soe notation, which is siilar to that used in [9]. The doain is decoposed into a finite nuber of control volues (CVs) using a grid. A grid is denoted by a letter such as h, which will also be interpreted as the distribution of the grid spacing in the doain. Therefore grid ah is such that it s spacing at each location equals a ties the spacing of grid h at the sae location. Given a grid h, such a process of obtaining grid a h will be referred to as systeatic refineent if a < 1, or systeatic unrefineent if a > 1. The set of all CVs of a grid h is denoted as V h. The set of all points in the region where the partial differential equation is defined will be denoted by Ω, and G(Ω) will denote the set of functions which are defined on Ω. Analogously Ω h Ω will denote the set of centroids of the CVs of grid h and G(Ω h) will denote the set of all functions defined on Ω h (grid functions). Also if G(Ω) then h G(Ω h) will denote the grid function such that h(x) = (x) for all x Ω h. Letters in bold italic such as x refer to position vectors in space. Also h, or ( h) is the -th coponent of the grid function h, that is the value of h at the centroid of CV. The operator which saples the function at the CV centroids to return the grid function h is given as I 0h :G(Ω) G(Ω h), i.e. I 0h = h h(x) = (x) for all x Ω h. Since grid functions are usually used to represent functions of continuous space one can define the inverse operator I h0 :G(Ω h) G(Ω) such that I h0 h = (x) = h(x) for all x Ω h. For the rest of the points x Ω h, (x) will assue a value deterined by a suitable interpolation, so I h 0 is not unique. By siilar reasoning a grid function ay be transferred fro a grid h, say, to a grid k by the operator I h k : G(Ω h) G(Ω k), I h k = I 0 k I h0. Again a suitable interpolation ust be chosen. Suppose a differential operator N which acts on a function and returns a function N. The finite volue ethod approxiates the integral of the operator iage over each CV by an algebraic expression. If is a CV covering a volue ΔΩ then the finite volue ethod starts by deriving a relation of the for: 1 N d Nhh h (2.1), where N h is an algebraic operator which approxiates the average of N over each CV and τ h is the truncation error associated with N h. The discretisation should be such that the truncation error tends to zero as h 0. In this case the left hand side of (2.1) tends to the value of N at the centroid of, and since τ h 0 so does (N h h) due to (2.1). The saller τ h, the better N h approxiates the average of N at CV. Equation (2.1) is written in a for which aids theoretical understanding but in practice finite volue ethods usually construct an algebraic expression which is equivalent to ΔΩN h in an effort to approxiate ΔΩN dω. To solve the differential equation N = b, where b is a known function, by the finite volue ethod, the equation is first integrated over each CV giving ΔΩN dω = ΔΩb dω for each V h, and then substituting (2.1) for the left hand side one obtains: N 1 b d (2.2), h h h The grid spacing need not be a physical spacing. For exaple, for structured grids it ay be defined as the spacing in the coputational doain (as opposed to the physical doain). What is iportant is that it be defined so that relations of the for (2.5) - (2.7) ay be derived. 2

3 Equation (2.2) is exact and so if one was able to solve it one would obtain h, the exact values of at the centroids of the CVs. Unfortunately, this is not possible since τ h is not known. Instead one akes the assuption that the truncation error is sall enough such that dropping it would not change the solution of the syste significantly. Thus instead of (2.2), the following syste is solved over each CV : N h h 1 b d (2.3) The solution h of the syste (2.3) is not the sae as the exact solution h. It differs by the discretisation error ε h = h h. As the grid is systeatically refined and h 0 the truncation error will tend to zero and the systes (2.2) and (2.3) will tend to becoe equivalent. Therefore as h 0 h h and ε h 0. An analytic expression for τ h can be derived as N h is constructed fro N using Taylor series. The truncation error for CV will be of the for: 1nb k h, ck, n hn (2.4) k p n1 for soe p 1, where nb is the nuber of neighbours of CV which participate in its finite difference stencil and h n is the characteristic size of each of these neighbours, including itself. The coefficients c k,n will be functions of the derivatives of in the vicinity of. If the grid is refined systeatically then the characteristic sizes of the neighbours of will be proportional to the characteristic size h of itself. In this case (2.4) ay be written as: k h, ck h (2.5) k p Through systeatic refineent the space originally occupied by CV will becoe occupied by ore CVs. However, as the refineent is systeatic the truncation error for these new CVs will be given by the sae forula as for. In addition if the derivatives of vary continuously and the grid spacing h is sall enough then the coefficients c k will not be very different for the new CVs than for the original CV. The change in the agnitude of the truncation error will therefore be ostly due to the reduction in grid spacing h. As systeatic refineent progresses the ters c kh k with k > p of (2.5) will eventually becoe negligible copared to the ter c ph p and the truncation error will be reduced alost proportionally to h p. The discretisation schee N h is characterised as p-th order accurate. For linear operators it can be shown that systeatic refineent causes the discretisation error to reduce at the sae rate as the truncation error. The sae has been deonstrated experientally for the Navier-Stokes operator by any researchers see e.g. [10], [11]. The above discussion is suarised by the following relations, which hold for a p-th order accurate discretisation schee, and which state that through systeatic refineent the truncation and discretisation errors, treated as functions of the continuous space Ω, retain their shape but their agnitude tends to becoe proportional to h p : 0 p h h I O h (2.6) I O h (2.7) 0 p h h 3

4 3. Truncation error estiate The truncation error estiator, which is the one used in [8] and [9], will now be briefly described. To estiate τ h on grid h, this estiator considers the sae discretisation schee on a systeatically coarser grid, say. If, for brevity, one defines the grid function b h whose -th coponent equals ( ΔΩb dω) / ΔΩ (i.e. the right-hand side of (2.2)), then on grid the relations which correspond to (2.2) and (2.3) are: Fro (2.6) and (2.7) it is deduced that: N b (3.1) I I N b (3.2) I (3.3) 0 2 p 0 2 h h h I (3.4) 0 2 p 0 2 h h h A siple estiate of the truncation error begins by trying to estiate τ. Equation (3.1) cannot be used to calculate τ because the exact solution is not known. However, since the solution h on grid h is ore accurate than, one ay use it to approxiate the exact solution. Therefore an estiate for τ coes by substituting I h h instead of in (3.1): h h h b N I (3.5) The quantity τ h = b N (I h h ) is called the relative truncation error of grid with respect to grid h (it is used in the context of FAS ultigrid ethods). It can be readily calculated given the two grids h and, and the solution h on the fine grid. It is also dependent on the restriction operator I h. Adding τ h to the left hand side of (3.2) akes the solution of this syste equal to the fine grid solution I h h, just like adding τ to the left hand side of (3.2) akes the solution of this syste equal to the exact solution. Since τ h is an approxiation to τ, one can use (3.3) to obtain an approxiation for τ h. However, a ore accurate estiate is possible. If N / ( ) is the jacobian atrix of the discrete operator N at and the grid is fine enough such that the differences between the functions, I h h and are sall enough then the following hold: N N N N (3.6) / / N I N N I N I (3.7) / / h h h h h h h The second approxiate relation of (3.6) derives fro the first one due to (3.1) and (3.2), and siilarly the second approxiate relation of (3.7) derives fro the first one due to the definition of τ h and (3.2). Using the fact that ε = in (3.6), and the fact that I h h = ( ) ( I h h ) ε I h ε h in (3.7), there result respectively: N / 2 I N (3.8) (3.9) / h h h h But (3.9) can change further by deducing fro (3.4) that I h ε h ε / 2 p so that ε I h ε h [(2 p 1) / 2 p ]ε. Therefore (3.9) gives: 4

5 p 2 1 / h N p (3.10) 2 Coparing (3.8) and (3.10) one gets: p h p (3.11) and using (3.3) one arrives at the final truncation error estiator: 1 I 2 1 h h h p (3.12) Suarising, to estiate the truncation error: first solve the syste on grid h, second restrict the solution to grid, third calculate the relative truncation error by (3.5), and finally apply (3.12). For this estiate to work it is crucial that the operator N is constructed using the sae discretisation schees as N h. Actually it is not necessary to use grid, any ultiple rh will do and (3.12) holds with r in place of 2. Also to ensure that the errors introduced in the restriction of the fine grid solution do not spoil the truncation error estiate it would be a good idea to use in (3.5) a restriction operator I h of order higher than the order p of the discretisation. This will ensure that as the grid is refined the error introduced by the restriction operator will eventually becoe negligible copared to the truncation error. Also it ust be stressed that in the presentation so far the discrete systes have been written so that they express quantities per unit volue. As has already been pointed out, finite volue ethods usually construct discrete systes which approxiate the total fluxes and forces on each CV. Therefore after restriction of the velocity and pressure obtained on the fine grid and application of the coarse Navier-Stokes operator one obtains the product ΔΩ τ h, for each coarse CV. This quantity ust be divided by the volue ΔΩ of each CV to obtain τ h before (3.12) can be applied. Finally, it is appropriate to discuss the iplications of the approxiate solution of the discrete systes by iterative solvers. Indeed, it is not possible in general to solve the discrete Navier-Stokes syste exactly, but the residual ay be ade as sall as desired, up to achine precision, by perforing an appropriate nuber of iterations. If h k is the approxiate solution to syste (2.3) after iteration k and r h k is the associated residual then: N b r (3.13) k k h h h h Subtracting (3.13) fro (2.2), and (2.3) fro (2.2) one gets respectively: N N r (3.14) k k h h h h h h N N (3.15) h h h h h Coparing (3.14) with (3.15) it is easy to see that the solution h k corresponds to a truncation error τ h r hk, just as solution h corresponds to the truncation error τ h. To attain the full accuracy that a finite volue ethod can offer the residual should be reduced to the level of the truncation error in every CV of the grid. Furtherore to accurately estiate the truncation error, the residual should be saller than the truncation error in every CV, say r h k, 0.1τ h, for every CV. Therefore, for convergence of the iterative ethod one should not onitor the ean residual but the residual / truncation error ratio in every CV. The residual acts as a source of algebraic error h h k, just as the truncation error acts as a source of discretisation error see [12]. Therefore, a high residual in one region ay generate a high algebraic error in another where the residual itself is sall. On the other hand there is no point in reducing the residual far 5

6 below the truncation error, as (3.14) and (3.15) indicate: The algebraic error h h k would reduce, but the exact error h h k would not. Again, usually the discrete systes of finite volue ethods are such that the quantity ΔΩ r h k, is ore easily attainable for each CV nd -order finite volue discretisation for the Navier Stokes equations Here the particular discretisation schees which will be used in section 6 to test the ethod are briefly described. The 2-D stationary incopressible Navier-Stokes equations under constant density ρ and viscosity μ, integrated over a CV of volue ΔΩ are written in cartesian coordinates as: 1 x Nh, u, v, p V nu ds u nds p i nds S S S 0 1 y Nh, u, v, p V nv ds v nds p j nds S S S 0 (4.1) (4.2) 1 Nh, u, v V nds 0 (4.3) where S is the surface of CV, n is the outward noral unit vector at each point of the surface, i and j are the unit vectors in the x- and y- directions, u and v are the coponents of the velocity vector V = ui + vj, and p is the pressure. The boundary of each CV will be coposed of a nuber of straight faces, each of which separates it fro another single CV or fro the exterior of the coputational doain. Figure 1 shows a face f separating two CVs, with centroids and N. The centre of the face is denoted by c, and c denotes the point on the line N which is closest to c. Also points and N are such that the segent N is of the sae length as N, and is perpendicular to the face f, and its idpoint is point c. The part of the grid shown in figure 1 exhibits skewness, that is the line joining and N does not pass through the centre c of face f. It is also non-orthogonal, which eans that the angle θ between N and the face noral is non-zero. Finally, if the iddle of the line segent N is far fro face f then the grid will also be said to exhibit expansion. The gradient operator is frequently used in discretisation schees and here it will be approxiated using the least squares ethod suggested in [4]. This ethod assigns to the discrete gradient h of the variable h at the centre of a CV the appropriate value so that the su Σ N{[Δ N ( h h) Δr N] / Δr N } 2 is iniised (the index N runs through all neighbours of, and Δ N = h,n h,, Δr N = N ). See [11] for an explicit expression for h in the twodiensional case. In the following, hx, h y will denote the two cartesian coponents of h. The Navier-Stokes equations will be discretised by approxiating the fluxes and forces on each CV face, using the sae or siilar schees as in [12]. In the following, an overbar denotes a value obtained by linear interpolation at point c fro the values at points and N, and a subscript c denotes a kind of linear interpolation, suggested in [12], which accounts for skewness and approxiates the value at point c as: S c c' (4.4) h, c h c' h h c' Also, the value of at point is approxiated as: 6

7 , ', ' (4.5) h h h h Then, the various ters of the x-oentu equation are discretised as: 1 1 x V n u ds Fh, f uh, c Ch, uh, vh, ph (4.6) S f f 1 1 uh, N' uh, ' x u nds S f Dh, uh (4.7) N' ' S f f 1 1 x x p in ds ph, cn f S f h, ph (4.8) S f f In the above, f is the set of all faces of CV and n f x, n f y are the cartesian coponents of the outward unit vector n f which is perpendicular to f. Also F h,f is the discrete ass flux through face f, to be defined shortly. The y-oentu equation is discretised analogously, while the continuity equation is discretised as: 1 1 V n ds Fh, f Nh, uh, vh, ph (4.9) S f f The su of the approxiate discrete operators (4.6)-(4.8) is the discrete x-oentu operator N h (u h,v h,p h) = C h (u h,v h,p h) D h (u h) h (p h), which tries to approxiate the exact x- oentu operator N h x (4.1). The associated truncation error with respect to the exact solution (u,v,p) of (4.1)-(4.3) is τ h x = N hx (u,v,p) N h (u h,v h,p h). Siilarly, the discrete y-oentu and continuity operators approxiate the exact operators (4.2) and (4.3) up to truncation errors τ h y = N hy (u,v,p) N h y (u h,v h,p h), τ h = N h (u,v) N h (u h,v h,p h). The truncation errors τ hx, τ hy, τ h will be estiated using (3.12). The above schees are in general considered to have truncation errors of O(h 2 ) see [12] so p = 2 will be used in (3.12). Most of the above discretisation schees use linear interpolation, which has the effect that the iage L h h of a discrete operator L h which uses it ay be sooth even if h contains a coponent which oscillates fro CV to CV (i.e. with period of oscillation equal to two CVs). Or equivalently, the solution h of the syste L h h = b h ay contain oscillations even if b h is sooth. A special case of this is the so-called checkerboard distribution depicted in figure 2: If, for exaple, the pressure at the CV centres assues the values shown in the figure, then obviously linear interpolation gives zero pressure at the face centres, and the operator h (4.8) gives zero pressure force on each CV. At doain boundaries pressure is extrapolated fro the interior, and such an oscillating pressure field would result in non-zero oscillating forces along the boundary CVs, which eans that an oscillating pressure field is not part of the null space of h. Therefore one ay be tepted to think that according to (3.15), as h 0 if τ h 0 then p h will tend to the exact pressure p h which is oscillations-free. However, the oscillating pressure field is close to being an eigenvector of h corresponding to a zero eigenvalue, and the saller the grid spacing h the closer it is to such an eigenvector. In practice this eans that pressure oscillations ay indeed appear in the discrete solution and they ay be very resistant to grid refineent. A siilar, but not as bad, situation holds also for the velocity field. The convection operators (4.6) and (4.9) produce iages which ay be sooth even if the velocity coponents oscillate at CV centres. However the discrete viscous force operator (4.7) involves direct velocity differences between adjacent CV centres and therefore always reflects velocity oscillations to its iage. Consequently, the phenoenon of oscillations in the u h, v h fields becoes less intense 7

8 as the Reynolds nuber decreases, and in fact oscillations diinish with grid refineent, and are rarely a proble for incopressible flows. The discrete gradient operator h ay also produce a sooth iage when applied to an oscillating field. ressure oscillations are a serious proble for colocated grids which has been addressed by any researchers. One way around it is to observe that the pressure forces are calculated fro values of pressure estiated at face centres using linear interpolation. These values have a uch saller discretisation error than the oscillating values at CV centres. So after obtaining the solution to the discrete syste one ay discard the pressures at the CV centres and consider the pressure field to be given by the pressures at the face centres. But rather than obtaining an oscillating pressure field and eliinating the oscillations afterwards it is ore desirable to obtain an oscillations-free field altogether, to avoid probles for the algebraic solvers of the discrete syste. One possibility is to use another discrete operator for the pressure force, one which reflects pressure oscillations to its iage, like the one proposed in [13]. However, the ost popular ethod involves adding an artificial pressure ter to the discrete expression for the ass flux through a face, generally known as oentu interpolation. Moentu interpolation was originally proposed in [14]. Since then any variants of this technique have been proposed but ost of the share the feature that the discretisation of the face ass fluxes is interlinked with a SIMLE-like solution ethod (there are a few exceptions, e.g. [10]). SIMLE-like solution algoriths linearise the oentu equations to obtain linear systes for the velocity coponents, whose -th equation has the for: (4.10) A, u, A, u, Q p, S n i u u u,\ p h N h N h c f f N ff where N runs over all neighbours of CV, and A u ij is the (i,j)-th coefficient of the atrix of coefficients of the linear syste for u. If face f separates CVs and N and n f points fro to N (see figure 3) then the oentu interpolation variant of [12], which is ore appropriate for our discretisation, approxiates the ass flux through f as: S f 1 Fh, f cs f Vh, c nf ai ph, ph, N h ph h ph N u N A f 2 (4.11) where: u 1 u u Af A, AN, N (4.12) 2 Here V h,c = u h,ci + v h,cj, and a i is a real factor introduced for better control of the pressure ter. Most researchers use a i = 1. Obviously (4.11) is equivalent to interpolation (4.4) but with the addition of a pressure ter. However, ass fluxes are functions of velocity only and therefore the pressure ter relates copletely to the truncation error. Therefore the agnitude of the pressure ter should diinish as h 0, at a rate which is at least 2 nd -order to preserve the overall order of accuracy of the discretisation. Indeed using Taylor series one can show that, if h is at least 2 nd -order accurate: 1 2 p 3 h, p h, N p h h p h h N O h N (4.13) In addition S f O(h) and A u f O(1), so the pressure-part of the discrete ass flux is: S a p p p p O h 2 f 1 ic u h h N h h h h N A f 2 5,, N (4.14) 8

9 By dividing by the volue ΔΩ O(h 2 ) of a CV which shares the face f one sees that (4.14) contributes a O(h 3 ) coponent to the truncation error. In fact, in [11] it is shown that under special but not uncoon circustances the su of the ters (4.14) for two opposite faces of a quadrilateral CV becoes O(h 6 ) because the leading ters of their Taylor expansions cancel out. This corresponds to a O(h 4 ) contribution to the truncation error. The pressure-part (4.14) of the discrete ass flux consists of the difference between two parts, one involving the direct pressure difference p h, p h,n between the centroids of the adjacent CVs, and one involving the pressure gradient. The part involving the pressure gradient is again insensitive to pressure oscillations, but the part involving the direct pressure difference is not. Indeed, if pressure oscillates fro one CV to the other then the pressure difference will also oscillate fro face to face, and so will the discrete ass flux. Therefore, the discrete Navier-Stokes and continuity operators do reflect pressure oscillations to their iage, which reoves pressure oscillations fro the discrete solution. The ain disadvantage of expression (4.11) is that to calculate the ass fluxes F h one needs the coefficients of the atrix A u, but to calculate the coefficients of A u one needs the ass fluxes F h! Therefore, given a discrete flow field u h, v h, p h one cannot directly evaluate the ass fluxes through the faces of the CVs but has to resort to an iterative procedure. For our truncation error estiator this eans that the expression (4.14) cannot readily be evaluated on the coarse grid. One ay argue that since the agnitude of the pressure ter of F h reduces at a rate which is faster than 2 nd -order then it ay siply be oitted on grid. On the other hand, including this ter on grid allows for a cleaner approach which ay also be used with up to 4 th -order accurate overall discretisation schees. Indeed, schees based on higher-order rather than linear interpolation ay also allow for oscillating pressure fields, and this is why in [15] oentu interpolation is used in the context of a 4 th -order accurate discretisation. 5. New oentu interpolation One idea to overcoe the above proble is to use in (4.11) only the viscous-part of A u which contains only geoetric ters and does not depend on F h. However, this leads to the coefficient of the pressure-ter of the ass flux being too big, resulting in divergence of the solution ethod unless a very sall value of a i is used (in [16] a i = 0.04 is used). This in turn was found not to eliinate the pressure oscillations at soe regions of the flow field. Therefore, the velocity field has to be taken into account, and this is done through A u. The reasoning behind the choice of the coefficient of the pressure ter is the following: Equation (4.10) suggests that the contribution of pressure to the value of u at point is: p 1 1 u p S n i p (5.1) x h, u h, c f f u h h A, f f A, where δ p u h, is the part of u h, which is due to pressure forces. The (discrete) Gauss theore is used to obtain the second equality. Of course the above assuption is very crude because in (4.10) the coefficients of A u are also functions of u h and v h, and u h,n also depend on the pressure field, and also upwind differencing is used to for A u while the central difference schee is iposed through deferred correction. However (5.1) gives a feel of the iportance of pressure in deterining u h. A siilar relation can be derived for v h, and in fact the coefficients of A v are nearly equal to the coefficients of A u, except aybe near soe boundaries. Therefore, in [12] the assuption is ade that a siilar relation holds for the coponent of velocity in any direction. In particular if u n h,c is the coponent of velocity noral to face f at c, u n h,c = V h,c n f, then it is assued that: 9

10 p n 1 uh, c h ph n u c f f (5.2) A f where the product ( hp h) cn f equals the pressure gradient in the direction of n f at the face centre. The volue ΔΩ f = S f(n )n f is defined as the volue of the iaginary CV around face f depicted by dashed line in figure 3, which has two sides parallel to face f and passing through points and N, and two sides perpendicular to f passing through its vertices. If one further approxiates ( hp h) c as the ean of ( hp h) and ( hp h) N then (5.2) becoes: S p n f 1 uh, c h ph h ph N u N A 2 f (5.3) Actually and N (see figure 3) should be used instead of and N in (5.3) but since the pressure contribution is very approxiate this substitution is acceptable. Because of (4.13) one can substitute (5.3) by: S,,, (5.4) p n f uh c p u h ph N Af By substituting the pressure contribution (5.3) to the noral coponent of velocity by (5.4) and using the idpoint rule that F h,f = ρ c S f u n h,c one arrives at (4.11). The above reasoning, although it does not sound very solid, in practice gives an appropriate agnitude to the coefficient of the pressure ter of the discrete ass fluxes, with a i 1. If a i is uch saller then the ethod fails to eliinate the oscillations, while if a i is uch larger then the syste is difficult to solve and divergence occurs on grids of reasonable fineness (of course if the grid is fine enough then the significance of the pressure ter will diinish no atter what the value of a i). Now, to uncouple the ass flux discretisation fro the iterative solution ethod, in the present work A u f will be substituted in (4.11) by a pseudo-coefficient A f which depends directly on the grid geoetry around face f and on the velocity at the adjacent CVs. Just as A u, is the coefficient by which u h, is ultiplied in the linearised discrete x-oentu equation of CV, A f is constructed as the coefficient by which u n h,c would be ultiplied in a hypothetical linearised discrete n-oentu equation for the iaginary CV around f in figure 3. The construction of the hypothetical oentu equation proceeds as follows. First define the points, NN, VV1, VV2 such that lies idway between c and etc. (see figure 3). Starting with viscous forces, for the side of the iaginary CV which passes through N, the n-coponent is discretised as: N n n n n uh NN uh c u n h NN uh c u ds, ',, ', n S f S NN' c 2 N' c f (5.5) where u n is the coponent of velocity noral to face f. A siilar schee will be used for the viscous coponent over the face through. For the face through V1 it will be assued that: u u n n n h, VV1 h, c u nds SV (5.6) 2 V 1 c V1 10

11 where S V = (N )n f is the length of each of the faces which are perpendicular to f. A siilar assuption is ade for the face which passes through V2. Therefore the total contribution of viscous forces to the coefficient of u n h,c is: A Sf Sf S S 2 N' c 2 ' c 2 V 1 c 2 V 2 c visc V V f (5.7) This can be siplified because 2 V1 c = 2 V2 c = S f. Also since one is only interested in an approxiate value for A f, (5.7) can be further siplified by assuing that c lies idway between N and so that 2 N c = 2 c = (N )n f = S V. Therefore (5.7) becoes: A visc f S f 2 SV S S V f (5.8) If ρ and/or μ are not constant, then the Navier-Stokes equations include other viscous force coponents as well, but usually in the SIMLE fraework they do not contribute to the atrices A u, A v. Therefore the viscous contribution (5.8) reains the sae. For convection, in the original SIMLE ethod the coefficients of A u are fored using the upwind difference schee while the central difference schee is enforced through deferred correction. Therefore, an upwind-like approach will be used for the convective part of A f. It is assued that the velocity at the centre of each face of the iaginary CV equals V h,c and that the ass flux through each face is given by the usual idpoint rule, F = V h,cns. Whatever the direction of V h,c ass will flow out of the iaginary CV only through two of the faces, one parallel to f and one perpendicular. Therefore, according to the upwind philosophy only these two faces will contribute to A f. If by rotating the vector x 90 o anticlockwise one gets the vector rot(x), then rot(n f) is the unit vector which is perpendicular to the faces which pass through V1 and V2, and the su of the convective contributions to A f is: A S V n S V rot n (5.9) convec f c f h, c f c V h, c f It is reinded that S V = (N )n f. In total the value of A f is therefore: A A A (5.10) convec visc f f f The proposed oentu interpolation is therefore to use (4.11) with A u f replaced by A f given by (5.10), (5.9) and (5.8). The new ethod also overcoes the well-known proble of the dependency of (4.11) on the underrelaxation factor of the solution ethod and the tie step for transient flows see [17], [18]. Before ending this section it should be entioned that at first there was an effort to use central differencing for the convective part of A f which resulted in: A 1 V, V, n S (5.11) 2 convec f h N h f f However, this did not work. It sees that (5.11) gives too sall a value for the convective part because for soothly varying fields the difference V h,n V h, will be sall. Therefore, A f will again be doinated by the viscous, geoetric ters and the probles entioned earlier will arise. This also highlights the iportance of using upwind differencing for convection for the construction of the atrix of coefficients of several solution ethods including SIMLE. 11

12 6. Testing of the ethod The ethod is tested on two cases with analytic solution, which allows direct coparison between the truncation error estiate and the actual truncation error: A particular lid-driven cavity proble, and the flow between concentric cylinders. 6.1 Lid-driven cavity Moentu interpolation We start with a few coents on the new variant of oentu interpolation. Extensive results will not be presented because it was observed that in general this variant offers nearly identical accuracy and rates of algebraic convergence (if SIMLE is used) as classic oentu interpolation. Here are presented soe results of applying the ethod to the siulation of the flow in a skew lid-driven cavity of side L = 1 with side walls inclined at 45 o, and the top lid oving at V lid=1 /s (see [19]), at Re=1000 (ρ=1 kg/ 3, μ=0.001 as). The pressure is fixed to zero at (x,y) = (0.5, 0.01), where the origin (x,y) = (0,0) is at the lower-left corner. This is not the lid-driven cavity proble with analytic solution which will be used in the next section to assess the truncation error estiator, but it allows testing of the oentu interpolation on grids which are not cartesian. The proble was solved using unifor grids, with three different schees for the ass fluxes: 1) linear interpolation (i0) i.e. with ter (4.14) copletely absent fro (4.11), 2) classic oentu interpolation (i1), and 3) new oentu interpolation (i2). Figure 4 shows the pressure distribution along the horizontal line passing through the centres of the CVs which lie iediately above the line y=h3/4, where H = Lsin45 o is the height of the cavity, on two grids of different density. If i0 is used then pressure oscillations appear in the solution, whose aplitude increases towards the interior of the doain, and which do not diinish with grid refineent. In this case the oscillating coponent of the pressure field is close to being an eigenvector corresponding to the zero eigenvalue, which eans that sall changes in the algebraic residual ay reflect large changes in the aplitude of the oscillations. To iniise the possibility that the oscillations are a product of insufficient residual reduction, iterations were continued until the residual was below 10 5 N/ 3 in each CV of the grid. Also the solution was obtained using two different initial estiates, one being the prolonged solution of the iediately coarser grid, and the other being the sooth solution obtained with oentu interpolation, but no significant differences were observed. If oentu interpolation is used then a sooth pressure field is obtained, and on the grid the results of the two variants of oentu interpolation are indistinguishable. ressure oscillations also have a detriental ipact on the convergence rate of the SIMLE algorith, which was used to solve the discrete systes. For exaple, to solve the proble on the 6464 grid up to the 10 5 residual, using the solution of the 3232 grid as initial estiate, iterations were required in the i0 case, as opposed to 485 and 479 iterations in cases i1 and i2 respectively (underrelaxation factors a u=0.8, a p=0.3, and a 2 nd pressure correction for grid non-orthogonality were used as suggested in [12]). If ultigrid is used things would be even worse as pressure oscillations which developed in coarse grids would be prolonged to the fine ones. Also it is very iportant to note that for the coefficients of the pressure-correction syste of the SIMLE algorith, (4.12) ust be used instead of (5.10), otherwise the ethod diverges. This ay sound strange since both forulae give siilar values at convergence, but they ay differ significantly at the first stages of iteration because (4.12) is coputed fro ass fluxes which are calculated after the pressure correction step, while (5.10) fro velocities obtained before this step. Finally, it was found that with i2 for extreely coarse grids it ay be necessary to use a i < 1 otherwise SIMLE ay diverge, unlike if i1 is used, because A f (5.10) is a little saller than A u f (4.12) (due to the fact that the velocity 12

13 underrelaxation factor is not taken into account in A f, unlike A u ). This holds also for ultigrid ethods which use very coarse grids Analytic solution To test the truncation error estiator we apply it to the lid-driven cavity proble of [20], which has an analytic solution: Fluid of ρ=1 kg/ 3, μ=0.001 as is enclosed in a square cavity, whose sides of length L=1 are aligned with the x- and y- axes. The top wall (lid) oves with a horizontal velocity u(x,1) = 16(x 4 2x 3 +x 2 ), and there exists a body force b in the y-direction: where: b(x,y) = 8μ[24F(x) + 2f / (x)g // (y) + f /// (x)g(y)] + 64[F 2(x)G 1(y) g(y)g / (y)f 1(x)] (6.1) f(x) = x 4 2x 3 + x 2 g(y) = y 4 y 2 F(x) = f(x)dx F 1(x) = f(x)f // (x) [f / (x)] 2 F 2(x) = f(x)f / (x)dx = 0.5[f(x)] 2 G 1(y) = g(y)g /// (y) g / (y)g // (y) where the pries denote differentiation. The exact solution to this proble (eqns. (4.1)-(4.3) with the addition of ( ΔΩbdΩ)/ΔΩ in the left-hand side of (4.2)) is: u(x,y) = 8f(x)g / (y) (6.2) v(x,y) = 8f / (x)g(y) (6.3) p(x,y) = 8μ[F(x)g /// (y) + f / (x)g / (y)] + 64F 2(x){g(y)g // (y) [g / (y)] 2 } (6.4) (actually any pressure field p = p + c will do, for any constant c). The proble is discretised using the schees of sections 4-5, plus the body force is discretised with the idpoint rule as ΔΩbdΩ b(x,y )ΔΩ. Since the exact solution is known, the exact truncation errors τ hx, τ hy, τ h of the operators N h, N h y, N h can be calculated. The calculation of τ h y requires integration of the body force over each CV, so we will focus on τ hx, τ h. The proble is solved on a unifor and a non-unifor series of structured grids of up to CVs. The non-unifor grids are such that, if each CV is assigned a horizontal index i and a vertical index j, and Δx h i,j, Δy h i,j are the horizontal and vertical sizes of CV (i,j) of grid h respectively, then there is a constant r h such that Δx h i+1,j / Δx h i,j = r h for x < 0.5 and Δx h i+1,j / Δx h i,j = 1/r h for x > 0.5, and siilarly for Δy. The expansion ratio r h for grid is such that the boundary CVs which touch the centrelines have a ratio of Δx/Δy = 10:1 or 1:10. Grid coes fro grid h by reoving every second grid line, so for the non-unifor grids r h = r and r h ranges fro about on grid 3232 (this grid is shown in figure 5) to about on grid Algebraic residuals were dropped below 10 8 in every CV. Figure 6 shows the exact distributions of τ h x and τ h on the grids. In the following, τ h, τ h y, τ h will denote the truncation error estiates (3.12), and ε hu, ε hv, ε h p the discretisation errors. Figures 7 and 8 (left) show the distributions of ε h u along the centres of the CVs which lie just to the right of the vertical centreline (x=0.5), and the distributions of ε h p (ε h p was set equal to zero at the centre of the CV of the lower left corner of the doain) along the centres of the CVs which lie just above the horizontal centreline (y=0.5). The errors are displayed in logarithic scale, and fro the distance between the distributions it is verified that the particular finite volue ethod is 2 nd -order accurate. Convergence is not as regular for the non-unifor grids as for the unifor ones, and in fact the discretisation errors on coarse nonunifor grids decrease at a rate which is faster than 2 nd -order, but tends to becoe 2 nd -order with refineent. For the calculation of τ h (3.5) two different restriction operators were used. The first is proposed in [12]: 13

14 1 (6.5) 4 C Ih h h, C hh C CC where C is the set of 4 CVs of grid h which cover CV of grid - the CVs of the set C will henceforth be called the children of the parent. The other is proposed in [11]: 1 4 I h h h, C h h C C CC xx x, C h C yy y h, C C xy h, C yx x h, C C y C 2 (6.6) where Δx C = x x C etc., and ( x ) h etc. are the approxiate second derivatives of on grid h, which are calculated by applying a least squares differentiation again to the coponents of h h. For the second differentiation, the neighbours of each CV C are considered to be its siblings (i.e. the CVs which have the sae parent as C). In [11] it is shown that (6.5) is 2 nd -order accurate as long as h is at least 1 st -order accurate, while (6.6) is 2 nd -order accurate if h is 1 st - order, and 3 rd -order if h is at least 2 nd -order. In our case h is 2 nd -order accurate except for the boundary CVs where it is 1 st -order. For prolongation in (3.12), the following 2 nd -order accurate operator was used: h I2 2 2, 2 2 C (6.7) h h C h h h The vertical centreline is not appropriate for the study of the truncation error because certain derivatives of the flow variables becoe zero there and this causes the leading ter of the truncation error to diinish, as shown in figure 6. This is shown clearly in figure 9, which shows τ h along the centres of the CVs just to the right of the vertical centreline, for a series of unifor grids: the distance between the distributions of consecutive grids indicates that τ h reduces alost at a 4 th -order rate. The estiate τ h is also plotted in the sae figure, and it is clear that it captures the overall shape of τ h but it is always about 4 ties higher. This is not surprising since the assuption that p=2 is ade in (3.12). In figure 10, τ hx is displayed along the CV centres just to the right of x=0.75 on the unifor grids. Here τ h x indeed reduces at a 2 nd -order rate. Two estiates τ h are also shown, one using (6.5) and one using (6.6). The one using (6.6) appears to be ore accurate, which is verified by the graph of the quantity (τ hx τ h )/τ hx. Since the estiate (3.12) is based on the assuption that the agnitude of the truncation error is deterined by its leading ter, the difference τ hx τ h should be of the order of the second leading ter of the truncation error, and therefore (τ hx τ h )/τ hx should be O(h) (except if only odd or even powers of h appear in the expansion of τ hx, in which case (τ hx τ h )/τ hx should be O(h 2 )). For the estiate using (6.5) this quantity reduces at a rate which is less than 1 st order while for the estiate using (6.6) it reduces at a rate which is faster than 1 st order (but less than 2 nd order). At y 0.32 and y 0.8 this quantity does not reduce, because the truncation error there reduces at a rate which is faster than 2 nd -order (actually it changes sign see figure 6). Therefore, a situation siilar to that shown in figure 9 occurs there. Siilar conclusions are drawn fro figure 10 concerning the distributions of τ h and τ h along the centres of the CVs just above y=0.75. Again the truncation error reduces at a 2 nd -order rate, and the quantity (τ h τ h )/τ h reduces at a rate which is a little faster than 1 st - order. In this case the estiate using (6.5) behaves as good as that using (6.6), and the two estiates are nearly indistinguishable. Figure 11 displays siilar data but for the non-unifor grids. This tie (τ hx τ h )/τ h x does not converge to zero if (6.5) is used, while if (6.6) is used then it decreases again at a rate which is just above 1 st -order. Oscillations which appear in the graph of (τ hx τ h )/τ h x have a period equal 14

15 to twice the grid spacing of the fine grid h, and so they ust be due to the prolongation operator (6.7). The behaviour of (6.5) is better for τ h than for τ h. Assuing that (τ h τ h )/τ h O(h), as nuerical experients confir when (6.6) is used, then τ h = τ h + τ ho(h) = τ h + O(h p+1 ), where τ h O(h p ). It follows that if instead of copletely dropping τ h in (2.2) one substitutes it by τ h, then the order of the approxiation would increase fro p to p+1. This is confired by nuerical experients: When τ h, τ h y, τ h were substituted instead of zero in the right hand sides of the discrete Navier-Stokes and continuity equations respectively, the discretisation errors reduced at a rate which is a little faster than 3 rd - order, dropping by nearly an order of agnitude per grid. This is deonstrated in figures 7 and 8 (right). The benefits appear saller for non-unifor grids, but the distance between the distributions shows 3 rd -order accuracy also in this case. This is verified also fro figure 12, which shows the agnitude of the approxiate integrals of the discretisation errors over the coputational doain. The penalty is that the equations ust be solved twice on each grid. Siilar investigations were conducted in [8]. 6.2 Flow between concentric cylinders Next we consider flow between two concentric cylinders, using a series of grids which exhibit non-orthogonality, expansion and skewness. The inner cylinder has a radius R 1 = 0.5 and is still, and the outer cylinder of radius R 2 = 1 rotates clockwise with a tangential velocity of V 2 = 1 /s (angular velocity ω 2 = 1 rad/s). If we use polar coordinates r, with the angle easured clockwise fro the vertical axis Oy, then the solution to this proble is a velocity field with a zero coponent in the r-direction and agnitude of V(r)=Ar+B/r where A=4/3, B= 1/3. The pressure, up to a constant, is given by p = ρ[a 2 r 2 /2 B 2 /(2r 2 ) + 2ABlnr] see [21]. The fluid has ρ = 1 kg/ 3, μ = 0.01 as (viscosity does not affect the solution, but it does affect the agnitude of the truncation error). Structured grids are used: One faily of grid lines consists of straight lines connecting the two cylinders, aking a 45 o angle with the radial direction at the inner cylinder (henceforth straight grid lines). The other faily consists of concentric circles, siilar to the cylinders theselves (henceforth circular grid lines). The finest grid has CVs and the circular grid lines lie at radial positions r j = cosθ j where θ j = π(j 1)/128, j=1,2,,129. The other grids coe fro reoving every second grid line fro the iediately finer grid. The 6416 grid is shown in figure 5. Figure 13 shows τ h x and τ h on the finest grid. The distribution of τ h y is the sae as τ hx, rotated by 90 o, and τ h is a function of r only. Again the results are of 2 nd -order accuracy as figure 14 shows. The behaviour of the truncation error estiates is siilar to the exaple of section 6.1.2: The estiate based on (6.5) converges to the exact error (in the sense that the ratio (τ h τ h )/τ h 0) at a rate which is slower than 1 st -order, or does not converge at all, while the estiate based on (6.6) converges faster than 1 st -order, nearly 2 nd -order in soe cases. This is deonstrated in figure 15. This tie, if one tries to increase the accuracy by adding τ h, τ h y, τ h to the right hand sides of the Navier-Stokes equations, no solution ay be obtained and the iterative procedure does not converge. In fact this is due to the odification of the continuity equation, which ay result in the syste having no solution. The odified discrete continuity equation for a CV is: Fh, f h, (6.8) f f Each ass flux F h,f through a face of also appears with opposite sign in the continuity equation of the neighbour CV which also owns face f, unless f is a boundary face. Therefore the su of the left hand sides of (6.8) over all CVs of the grid equals the su of ass fluxes through the boundary faces, because the ass fluxes through interior faces cancel out. This is zero for the present proble because all boundaries are solid walls. Therefore the su of the 15

16 right hand sides of (6.8) should also be zero, but this is not guaranteed by the present ethod as described so far. The su τ h,ρδω equals zero because the truncation error of the approxiation of the ass flux through a face f contributes with opposite sign to the continuity truncation errors of the CVs on either side of f. Also, the su τ h,,ρ ΔΩ over all CVs of the coarse grid is zero because the ass fluxes F which are calculated fro the restricted velocity and pressure fields contribute with opposite sign to the relative truncation errors of the CVs on either side of the face. This property is lost in converting the relative truncation error on grid into a truncation error estiate on grid h by (3.12) - (6.7). Replacing the prolongation operator (6.7) by the operator (I h ) C =, does not necessarily correct the proble because in the presence of grid skewness the parent CV of grid ay not cover the sae area as its children of grid h. However the proble ay be overcoe by a siple odification of the prolongation operator: If is a CV of grid and C C where C is the set of the children of then (I h τ h, ) C is ultiplied by a function a τ defined by: which ensures that: CVh a h h, C C h, C C p CV 2 1 h I a h,, h h, I C C CC 1 h h, 1 h, p a I2 h C p, 2 1 C V2 C 2 1 h C V 0 (6.9) where the first equality coes fro (3.12) with I h replaced by a τ I h, and the third equality coes fro (6.9). On sooth structured grids (i.e. which are constructed fro distributions of diensionless variables ξ, η which have continuous derivatives) skewness and expansion tend to zero with grid refineent, so a τ 1, and in fact it does so quite rapidly as figure 14 (right) shows. The prolongation operator need not be odified for τ h, τ h y. With this odification it was possible to solve the syste and to obtain a solution of higher yet still 2 rd -order accuracy, as deonstrated in figure 14 (left). We were not able to propose a definite explanation of why 3 rd - order accuracy is not achieved, but it is likely that this is due to the boundary conditions. Indeed in [11] it is shown that siple boundary conditions such as those used for the present probles, which are siilar to those proposed in [12], result in τ hx, τ h y O(1) at the boundary CVs. This can be seen also in figure 15, for τ h y at r = Conclusions With the aid of a new oentu interpolation, a truncation error estiate has been ipleented and tested on sooth structured grids which exhibit non-orthogonality, skewness and stretching. Under these conditions the estiate converges to the exact truncation error in the sense that (τ h τ h )/τ h O(h), provided that it uses a restriction operator of sufficient accuracy. In this case the estiate ay be used to increase the approxiation order of the discrete syste, if the boundary conditions are chosen appropriately. 16

17 REFERENCES 1. Hoffan JD. Relationship between the truncation errors of centered finite-difference approxiations on unifor and nonunifor eshes. J. Cop. hys. 1982; 46: Lee D, Tsuei YM. A forula for estiation of truncation errors of convection ters in a curvilinear coordinate syste. J. Cop. hys. 1992; 98: Mastin CW. Truncation error on structured grids. In Handbook of Grid Generation, Thopson JF, Soni BK, Weatherill N (eds.); CRC ress, 1999; pp (32-1) (32-10) 4. Muzaferija S. Adaptive Finite Volue Method for Flow rediction using Unstructured Meshes and Multigrid Approach. h.d. thesis, Iperial College, University of London, Jasak H. Error Analysis and Estiation for the Finite Volue Method with Applications to Fluid Flows. h.d. thesis, Iperial College, University of London, Habashi WG, Dopierre J, Bourgault Y, Ait-Ali-Yahia D, Fortin M, Vallet MG. Anisotropic esh adaptation: towards user-independent, esh-independent and solver independent CFD. art I: general principles. Int. J. Nuer. Meth. Fluids 2000; 32: Brandt A. Multi-level adaptive solutions to boundary-value probles. Math. Coput. 1977; 31: Bernert K. τ-extrapolation - theoretical foundation, nuerical experient, and application to Navier - Stokes equations. SIAM J. Sci. Coput. 1997; 18: Trottenberg U, Oosterlee C, Schuller A. Multigrid. Acadeic ress: London, Deng GB, iquet J, Queutey, Visonneau M. Incopressible flow calculations with a consistent physical interpolation finite volue approach. Coputers & Fluids 1994; 23: Syrakos A. Analysis of a Finite Volue ethod for the incopressible Navier-Stokes equations. h.d. thesis, Aristotle University of Thessaloniki, Ferziger JH, eric M. Coputational Methods for Fluid Dynaics 2nd ed. Springer-Verlag: Berlin, Date AW. Solution of Navier-Stokes equations on non-staggered grid. Int. J. Heat Mass Transfer 1993; 36: Rhie CM, Chow WL. Nuerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA J. 1983; 21: Lilek Z, eric M. A fourth-order finite volue ethod with colocated variable arrangeent. Coputers & Fluids 1995; 24: Barton IE, Kirby R. Finite difference schee for the solution of fluid flow probles on non-staggered grids. Int. J. Nuer. Meth. Fluids 2000; 33: Majudar S. Role of underrelaxation in oentu interpolation for calculation of flow with nonstaggered grids. Nuerical Heat Transfer 1988; 13: Yu B, Kawaguchi Y, Tao WQ, Ozoe H. Checkerboard pressure predictions due to the underrelaxation factor and tie step size for a nonstaggered grid with oentu interpolation ethod. Nuerical Heat Transfer art B 2002; 41: Deirdzic I, Lilek Z, eric M. Fluid flow and heat transfer test probles for non-orthogonal grids: bench-ark solutions. Int. J. Nuer. Methods Fluids 1992; 15: Shih TM, Tan CH, Hwang BC. Effects of grid staggering on nuerical schees. Int. J. Nuer. Meth. Fluids 1989; 9: Warsi ZUA. Fluid Dynaics, Theoretical and Coputational Approaches 2 nd ed. CRC press:

18 FIGURES Figure 1: Geoetry around a face f separating control volues and N. Figure 2: Checkerboard distribution of a variable on a Cartesian grid. 18

19 Figure 3: Iaginary control volue around a face and related notation. Figure 4: 45 o skew cavity, Re=1000: ressure distribution along the CV centres just above the line y=3h/4 (H = height of cavity), on grids 3232 (left) and (right). 19

20 Figure 5: Left: the 3232 CV non-unifor grid for the lid-driven cavity proble. Right: the 6416 CV grid for the concentric cylinders proble. 20

21 Figure 6: The distributions of -τ h x (left) and -τ h (right) on the unifor grid (top) and nonunifor grid (below). 21

22 Figure 7: Top: Distributions of ε h u along CV centres just to the right of x=0.5, of the solution of the discrete Navier-Stokes syste whose right hand side equals zero (left in red), or equals inus the truncation error estiate (right in blue), on various unifor grids. Below: Siilarly for ε h p along CV centres just above y =

23 Figure 8: Like figure 7, but for solutions on non-unifor grids. 23

24 Figure 9: Absolute value of τ h (black) and τ h (colour) along the CV centres just to the right of x = 0.5, for various unifor grids. 24

25 Figure 10: Left side: The top diagra shows the distributions of τ h x (black) and τ h (purple: using (6.5), cyan: using (6.6)) at the CV centres just to the right of x=0.75, on various unifor grids, and the botto diagra shows the corresponding quantities (τ h x τ h )/τ h x. Right side: Siilar to the left side, but for τ h, τ h at CV centres just above y =

26 Figure 11: Left side: Top: τ h x (black) and τ h (purple: (6.5), cyan: (6.6)) at CV centres just to the right of the i=4 grid line of the 44 CV non-unifor grid (x0.88), on various non-unifor grids; botto: the corresponding quantities (τ h x τ h )/τ h x. Right side: Siilar to the left side, but for τ h, τ h at CV centres just above the j=4 grid line of the 44 CV non-unifor grid (y0.88). 26

27 Figure 12: Σ ε u h, ΔΩ, Σ ε v h, ΔΩ, Σ ε p h, ΔΩ of the solution of the original syste (red lines) and of the odified syste with τ h, τ h y, τ h added to the right hand side (blue lines). The left diagra refers to the series of unifor grids (grid 4 = 3232 CVs, grid 7 = CVs), and the right diagra to the series of non-unifor grids. Figure 13: -τ h x (left) and -τ h (right) on the grid. 27