NASA Technical Reports Server (NTRS) 20110011294: Notes on Accuracy of Finite-Volume Discretization Schemes on Irregular Grids

0.23 MB·English

by  NASA Technical Reports Server (NTRS)

NOTES ON ACCURACY OF FINITE-VOLUME DISCRETIZATION
SCHEMES ON IRREGULAR GRIDS
BORIS DISKIN(cid:3) AND JAMES L. THOMASy
Abstract. Truncation-error analysis is a reliable tool in predicting convergence rates of dis-
cretization errors on regular smooth grids. However, it is often misleading in application to (cid:12)nite-
volume discretization schemes on irregular (e.g., unstructured) grids. Convergence of truncation
errorsseverelydegradesongeneralirregulargrids;adesign-orderconvergence canbeachieved only
ongridswithacertaindegreeofgeometricregularity. Suchdegradation oftruncation-errorconver-
gence does not necessarily imply a lower-orderconvergence of discretization errors. In these notes,
irregular-gridcomputations demonstrate that the design-order discretization-errorconvergence can
beachievedevenwhentruncationerrorsexhibitalower-orderconvergenceor,insomecases,donot
convergeatall.
1. Introduction. These notes are a response to the recently published arti-
cle [18]. The article applies a truncation-erroranalysis to evaluate accuracy of (cid:12)nite-
volume discretization (FVD) schemes on general unstructured grids. The analysis is
accompanied by computations performed on regular and irregular grids. While we
agree with the analysis and its conclusions in application to regular smooth grids,
we considerthe truncation-erroranalysisandsome of the conclusionsderivedfrom it
invalid in application to irregular-gridcomputations.
On regular grids, convergence of truncation errors (also referred as local errors
in the literature) is an accurateindicator of convergenceof discretization errors(also
referredasglobal errors). However,the truncation-errorconvergenceisoftenmislead-
ing forFVD schemesde(cid:12)ned onirregular(e.g., unstructured) grids. As shownin [18]
and before this in [17], the second-order convergence of truncation errors for some
commonly used FVD schemes can be achieved only on grids with a certain degree of
geometric regularity. Other studies, e.g., [3, 6, 7, 10, 13, 14, 15, 16, 20, 21], showed
that truncation-errorconvergencedegradationon irregulargrids does not necessarily
imply a degradation of discretization-error convergence. Examples shown in the fol-
lowing sections con(cid:12)rm that on irregular grids, the design-order discretization-error
convergence can be achieved even when truncation errors exhibit a lower-order con-
vergence or, in some cases, do not converge at all. Note that these results do not
contradictthe Laxtheorem,whichstatesthatconsistency(convergenceof truncation
errors) and stability are su(cid:14)cient (not necessary) for convergence of discretization
errors. In fact, for some formally inconsistent discretization schemes, it has been
rigorouslyproved that the discretization errorsconverge[3, 6, 7, 13, 16, 21].
Article [18] applied a truncation error analysis to FVD schemes for the Poisson
equation. A thin-layer approximation was analyzed. It was shown that the trun-
cation error is O(1) (i.e., does not converge) in grid re(cid:12)nement unless the grids are
regular. Thediscretizationerroroftheschemewasinferredtobenon-convergent. By
coincidence, the particularthin-layerFVD scheme considered in [18] is indeed zeroth
order accurate on general unstructured grids. The inaccuracy is caused by poor ap-
proximation to the normal gradients with the thin-layer approximation, not by O(1)
convergenceof truncation errors.
(cid:3)National Institute of Aerospace (NIA), 100 Exploration Way., Hampton, VA 23681
([e-mail])alsoMechanical&AerospaceEngineeringDepartment,UniversityofVirginia,
Charlottesville,VA22902.
yComputationalAerosciencesBranch,NASALangleyResearchCenter,MailStop128,Hampton,
VA23681-2199 ([e-mail]).
1
In[18],ageneralconclusionwasdrawnthat\acompact(cid:12)nitevolumeapproxima-
tionoftheLaplacianhastorelyonsymmetriesinthegridtobe(cid:12)rst-orderaccurate."
This conclusion is incorrect. For example, a common (cid:12)nite-volume scheme equiva-
lent to a Galerkin (cid:12)nite-element approximation on triangles satis(cid:12)es the de(cid:12)nition
of a compact scheme and is known to have second-order discretization errors (and
zeroth order truncation errors) on irregular (non-symmetric) grids. We show addi-
tional computations on general mixed-element grids demonstrating the second-order
convergenceofdiscretizationerrorsandzerothorderconvergenceoftruncationerrors.
Article [18] also considered a central FVD scheme for an advection equation on
mixed-elementandperturbedquadrilateralgrids. Truncation-erroranalysisshoweda
zeroth-orderconvergenceinthe L1 norm. Supportingcomputationsshowedazeroth
orderconvergenceofdiscretizationerrors. ItwasconcludedthatFVDschemesforan
advectionequationarenon-convergentonnon-smoothirregulargrids. Theconclusion
is wrong because there are counter examples of FVD schemes with truncation errors
that do not converge on general irregular grids but with discrete solutions that con-
verge with at least (cid:12)rst order in any norm. The numerical scheme considered in [18]
is not representative of current practice | the central scheme is known to exhibit
erratic convergence of discrete solutions because of checkerboard instabilities. Tests
exposing the erratic convergence of discretization errors of the central scheme are
shown in Section 4 of these notes.
In the following Section 2, we provide de(cid:12)nitions and discuss relations between
truncation and discretization errors. Section 3 presents simple one-dimensional ex-
amples that expose shortcomings of the truncation-error analysis for irregular-grid
FVD schemes. Section 4 presents numerical two-dimensional computations for the
Poissonequationonmixed-elementgridsandforanadvectionequationonperturbed
quadrilateral grids similar to those used in [18].
2. Formulation. For completeness of the presentation, we start from a brief
formulation of (cid:12)nite-volume discretization (FVD) schemes and de(cid:12)nitions of the cor-
responding truncation and discretization errors. The FVD schemes are derived from
the integral form of a steady conservation law
(F(cid:1)n^) d(cid:0)= fd(cid:10); (2.1)
I ZZ
(cid:0) (cid:10)
wheref isaforcingfunction,(cid:10)isacontrolvolumewithboundary(cid:0),n^istheoutward
unit normalvector,andF is the (cid:13)ux vectorthat mayinclude viscousand/orinviscid
contributions. ThegeneralFVDapproachrequirespartitioningthe domainintoaset
ofnon-overlappingcontrolvolumesandimplementingnumericallyequation(2.1)over
each control volume. In this paper, we are considering node-centered FVD schemes,
where solution values are stored at the mesh nodes.
For two-dimensional (2D) node-centered FVD schemes, a median-dual partition
is constructed by connecting the mass centers of the primal-mesh cells with the mid-
points of the surrounding edges (Figure 2.1). These non-overlappingcontrol volumes
covertheentirecomputationaldomainandcomposeameshthatisdualtotheprimal
mesh. The discretization is applied at a sequence of re(cid:12)ned grids composed of trian-
gularandquadrilateralcellssatisfyingthe consistent re(cid:12)nement property [8,20]. The
propertyrequiresthemaximumdistanceacrossthecellstodecreaseconsistentlywith
increase of the total number of grid points, N. In particular, the maximum distance
should tend to zero as N(cid:0)1=2 in 2D computations.
2
P
1
P
2
PP
00
P
P 4
3
Fig.2.1. Median-dualpartitionfornode-centered(cid:12)nite-volumediscretizations. P0(cid:0)P4 denote
grid nodes. Thecontrol volume (dual cell)around P0 isshaded.
The main accuracy measure of any FVD scheme is the discretization error, E ,
d
de(cid:12)ned as the di(cid:11)erence between the exact discrete solution, Qh, of the discretized
equations (2.1) and the exact continuous solution, Q, to the di(cid:11)erential conservation
law
rF=f; (2.2)
E =Q(cid:0)Qh; (2.3)
d
Q is sampled at grid nodes.
Truncation error, E , measures the accuracy of the discrete approximation to
t
the di(cid:11)erential equations (2.2). For FVD schemes, the traditional truncation error
is usually de(cid:12)ned from the time-dependent standpoint [19, 22]. In the steady-state
limit, it is de(cid:12)ned (e.g., in [10]) as the residual computed after substituting Q into
the normalized discrete equations (2.1),
1
Et = 2(cid:0) fhd(cid:10)+ Fh(Q)(cid:1)n^ d(cid:0)3; (2.4)
j(cid:10)j ZZ I
(cid:0) (cid:1)
4 (cid:10) (cid:0) 5
where Fh is a reconstruction of the (cid:13)ux F at the boundary (cid:0), j(cid:10)j is the measure of
the control volume,
j(cid:10)j= d(cid:10); (2.5)
ZZ
(cid:10)
fh isanapproximationoftheforcingfunctionf on(cid:10),andtheintegralsarecomputed
according to some quadrature formulas.
For a given FVD scheme, we de(cid:12)ne a residual function
1
R(q)= 2 Fh(q)(cid:1)n^ d(cid:0)(cid:0) fhd(cid:10)3; (2.6)
j(cid:10)j I ZZ
(cid:0) (cid:1)
4(cid:0) (cid:10) 5
3
where q is a discrete function de(cid:12)ned at the grid nodes. The residual accounts for
interior discretization and all boundary conditions. Note that by de(cid:12)nition
R(Qh)=0 (2.7)
and
R(Q)=E : (2.8)
t
Substituting (2.3) into (2.7),
R(Q(cid:0)E )=0; (2.9)
d
and assuming the discretization error to be small comparing to the exact continuous
solution, Q, (jE j<<jQj), we obtain
d
E (cid:0)J(Q)E (cid:25)0; (2.10)
t d
where J(Q) is the Jacobian of R(q) computed for q =Q,
1 @
h
J(Q)= 2 F (Q)(cid:1)n^ d(cid:0)3: (2.11)
j(cid:10)j@q I
(cid:0) (cid:1)
4(cid:0) 5
Thus, the discretization error can be estimated as
E (cid:25)J(cid:0)1E ; (2.12)
d t
where the inverse Jacobian is a convolution operator with a kernel that usually pos-
sesses certain smoothness properties. The discretization error in a point can be ap-
proximated as a weighted sum of the undivided truncation errors,
E (cid:25) w (E ) j(cid:10) j; (2.13)
d j t j j
Xj
wheresummationindexj goesoverallcontrolvolumesandweightingfactors,w ,are
j
determined by the convolution kernel.
Two norms are used in this paper to evaluate magnitudes of the discrete errors:
the L1 norm (also referred as max-norm) is computed as the maximum absolute
value of a discrete function; the L norm is computed as the mean absolute value,
1
i.e., the sum of the absolute values divided by the number of discrete entries. Thus,
either norm of a constant discrete function is the same constant.
Irregular grid computations, in which convergence of discretization errors sur-
passes the convergenceof truncation errors,were reported and analyzed in literature
sincethe1960s(see,e.g.,[3,6,7,10,13,14,15,20,21]). In[13],discretizationschemes
demonstratingsuchconvergencewerecalledsupra-convergent. Whilearigorousproof
of discretization error convergence for FVD schemes on general irregular grids is not
yet available, there are several recent publications addressing supra-convergence on
irregulargrids. Papers[7,16]considerformallyinconsistent(notruncation-errorcon-
vergence) schemes for advection equations in 1D and 2D and prove convergence of
discretization errors; Barbeiro [3] proves second-order convergence of discretization
errorsfor \inconsistent" discretizations of 2D elliptic equations on nonuniform grids.
4
A qualitative explanation of supra-convergence observed for FVD schemes on
irregular grids was suggested by Giles [10]. Giles’ arguments can be illustrated as
follows. First consider a node-centered FVD scheme on a sequence of regular grids.
On such grids, truncation and discretization errors would converge with the same
design order. Now consider e(cid:11)ects of a local irregularity introduced, for example, by
shiftingonenode. Thenodeshiftleadstochangesinfaces(andcorresponding(cid:13)uxes)
ofthecontrolvolumecontainingthisnodeanditsneighboringcontrolvolumes. Fluxes
outside of this immediate neighborhood remain unchanged. Each (changed) (cid:13)ux at
an internal face contributes equal and opposite amounts to the undivided truncation
errorsonbothsidesoftheface. Thus,thechangeintroducedintothelocaltruncation
errorcanbe large(the changeina(cid:13)uxdivided bythe controlvolume), but the mean
value (and, often, higher moments) of the changes introduced into the undivided
truncation errors is zero. If the convolution kernel is su(cid:14)ciently smooth, e.g., for all
nodes outside of the immediate neighborhood of the perturbed node, the weighting
factors do not vary too much over the control volumes with perturbed truncation
errors,thenthediscretizationerrorchangesareclosetozerooutsideofthisimmediate
neighborhood. Even within this neighborhood, the discretization error changes are
small comparing with the changes in the truncation errors because of the averaging
e(cid:11)ect.
3. One-dimensional examples. Examples in this section show that common
FVDschemesonone-dimensional(1D)irregulargridsexhibitsupra-convergence. Sim-
ilar computations have been demonstrated in [15, 16].
3.1. Spatialdiscretizationgrids. A1Ddiscretizationgridisde(cid:12)nedasacom-
bination of the primal and dual nodes. The solution values are located at the primal
nodes; the (cid:13)uxes are located at the dual nodes. For node-centered discretizations,
a natural strategy is to place the primal mesh (cid:12)rst and, then, use this mesh as a
reference for placing dual control volumes.
The discretization grids used in this section are designed to demonstrate e(cid:11)ects
of grid irregularitiesand are described as follows. The (cid:12)rst and the last of the N +1
primal nodes, x ;i=0;1;:::;N, arealwayslocated at the ends of the computational
i
interval; the interior nodes can be distributed either uniformly or randomly. Either
distributionretainsthenodalorderingandensuresthatthemaximaldistancebetween
theneighboringnodesisO(1=N). Lets ;i=0;1;:::;(N+1)denotethe(cid:13)uxlocations.
i
The (cid:12)rst and the last (cid:13)uxes arealso located at the ends of the interval. The location
of an interior (cid:13)ux, si, is always between the primal nodes xi(cid:0)1 and xi and initially
de(cid:12)nedasthe midpoint;the dualnodes maythen berandomlyperturbedaboutthe
i
midpoint.
Speci(cid:12)cally, on an interval x 2 [a; b], the primal nodes are distributed according
to
b(cid:0)a
x =a; x =a+(i+r ) ; i=1;:::;(N (cid:0)1); x =b; (3.1)
0 i i N
N
where r is either zero (uniform primal mesh) or a random number (cid:0)0:4 (cid:20) r (cid:20) 0:4
i i
(random primal mesh). The dual-mesh nodes are computed accordingly as
s0 =a; si =xi(cid:0)1+di(xi(cid:0)xi(cid:0)1); i=1;:::;N; sN+1 =b; (3.2)
where d = 0:5 and d = 0:5+rs correspond to unperturbed and perturbed dual
i i i
meshes, respectively; here rs is a random number (cid:0)0:25(cid:20)rs (cid:20)0:25. Grid examples
i i
5
(a)Uniformprimalmesh;unperturbeddualmesh.
(b)Uniformprimalmesh;perturbeddualmesh.
(c)Randomprimalmesh;unperturbeddualmesh.
Fig. 3.1. Examples of one-dimensional discretization grids: black bullets denote primal mesh
nodes, verticaltic-marksdenote dual mesh nodes.
are shown in Figure 3.1. The computations in this section refer to tests performed
on the interval x 2 [0; 1] using a sequence of grids with the total number of primal
gridnodesincreasingasN =2n+1; n=3;4;:::;14. All random gridsaregenerated
independently,sothereisnoregularityinthelimitofgridre(cid:12)nement. Theequivalent
mesh size is taken as h= 1.
N
100
10−2
10−4
10−6
10−8
Discr. Err. Max−Norm
10−10 Discr. Err. L1−Norm
Trunc. Err. Max−Norm
Trunc. Err. L1−Norm
10−12 3rd−order slope
2nd−order slope
1st−order slope
10−14
10−5 10−4 10−3 10−2 10−1 100
Equivalent Mesh Size
Fig. 3.2. Convergence ofdiscretization and truncation errors ingrid-re(cid:12)nementcomputations
fortheconstant-coe(cid:14)cientconvectionequation. Thetestsareperformedwithrandomprimalmeshes
and unperturbed dual meshes.
6
3.2. Steady convection equation. The (cid:12)rst example is a steady constant-
coe(cid:14)cient convection equation
@ U =f(x); U(0) = ; (3.3)
x
which is satis(cid:12)ed with the exact solution U =sin(x) ; f =cos(x). FVD equations
are formed as follows
F (cid:0)F =sin(s )(cid:0)sin(s ); i=0; :::;N: (3.4)
si+1 si i+1 i
The(cid:13)uxes,F ,approximatesolutionvaluesatthe(cid:13)uxlocations,s ,andarecomputed
si i
byfully-upwindextrapolationsfromtheprimalnodes(exceptforthe(cid:12)rstinteriordual
node) as
F = ;
s0
FFs1 == ((ssi1(cid:0)(cid:0)xxi0(cid:0))2Ux)11U+(cid:0)i(cid:0)(xx101(cid:0)(cid:0)(ss1i(cid:0))Ux0i(cid:0);1)Ui(cid:0)2; i=2;:::;N +1 (3.5)
si xi(cid:0)1(cid:0)xi(cid:0)2
whereU isadiscreteapproximationtoU(x ). Boundaryconditionsareimplemented
i i
weakly by de(cid:12)ning the in(cid:13)ow (cid:13)ux. The forcing term is integrated exactly over the
dual volume.
Figure 3.2 shows convergence rates obtained in grid-re(cid:12)nement on irregular dis-
cretization grids involving random primal meshes and unperturbed dual meshes. In
eithertheL1 orL1 norm,truncationanddiscretizationerrorsconvergewithdi(cid:11)erent
orders.
Primal Dual Grid-re(cid:12)nement computations
Mesh Mesh Discr. Error Trunc. Error
Uniform Unperturbed O(h2) O(h2)
Uniform Perturbed O(h2) O(h)
Random Unperturbed O(h2) O(h)
Random Perturbed O(h2) O(h)
Table 3.1
Convergenceofdiscretizationandtruncationerrorsfortheconstant-coe(cid:14)cientconvectionequa-
tion.
Table 3.1 summarizes discretization and truncation error convergence rates ob-
servedincomputations. Theresultsdemonstratethatgridirregularitystronglya(cid:11)ects
the truncation-error convergence, but has no e(cid:11)ect on the convergence order of the
discretization errors.
3.3. Time dependent convection equation. The time-dependent constant-
coe(cid:14)cient convection equation is de(cid:12)ned on the domain (t;x)2[0;1](cid:2)[0;1] as
@ U +@ U =f(x); U(t;0) = (t); U(0;x) =(cid:30)(x); (3.6)
t x
and is satis(cid:12)ed with the exact solution U(t;x) = sin(2(cid:25)(x(cid:0)t)); f (cid:17) 0. In com-
putations, the number of time levels is equal to the number of spatial grid nodes,
N +1. The time step is constant, h = 1=N: The same spatial grids are used at all
t
time levels.
FVD equations at time level k are formed as follows
(cid:0)fk+@hUk (s (cid:0)s )+Fk (cid:0)Fk =0; i=0; :::;N: (3.7)
i t i i+1 i si+1 si
(cid:0) (cid:1) 7
Here, @hUk is a discrete time derivative and fk is a discrete approximation to the
t i i
forcing term; both are de(cid:12)ned at grid node x . For k > 1, @hUk is the second-order
i t i
backward di(cid:11)erence scheme,
1 3 1
@hUk (cid:17) Uk(cid:0)2Uk(cid:0)1+ Uk(cid:0)2 ; (3.8)
t i h (cid:18)2 i i 2 i (cid:19)
t
and, for k = 1, the time derivative is approximated by the (cid:12)rst-order backward
di(cid:11)erence scheme,
1
@hU1 (cid:17) U1(cid:0)U0 : (3.9)
t i h i i
t (cid:0) (cid:1)
Thespatial(cid:13)uxes,Fk,areformedaccordingto(3.5). Thetimedependenttruncation
si
error is evaluated for the exact solution, U, at each dual volume as
Fk (cid:0)Fk
E =(cid:0)fk+@hU + si+1 si: (3.10)
t i t s (cid:0)s
i+1 i
101
100
10−1
10−2
10−3
10−4
Discr. Err. Max−Norm
10−5 Discr. Err. L1−Norm
Trunc. Err. Max−Norm
Trunc. Err. L1−Norm
10−6 2nd−order slope
1st−order slope
10−7
10−4 10−3 10−2 10−1 100
Equivalent Mesh Size
Fig. 3.3. Convergence ofdiscretization and truncation errors ingrid-re(cid:12)nementcomputations
for the time-dependent constant-coe(cid:14)cient convection equation. The errors are computed at the
(cid:12)naltimet=1. Thetestsareperformed withrandom primalmeshesandunperturbed dualmeshes.
Figure 3.3 shows convergencerates obtained in successivegrid-re(cid:12)nement, which
doubles the number of primal mesh intervals and time steps. Spatial grids involve
random primal meshes and unperturbed dual meshes. The coarsest mesh includes 9
primalnodes(8intervals)anduses9time levels(8steps). Truncationanddiscretiza-
tion errors are computed at the (cid:12)nal time t = 1. In either the L1 or L1 norm, the
truncationerrorsconvergewith the(cid:12)rstorderanddiscretizationerrorsconvergewith
the second order.
8
3.4. Di(cid:11)usion equation. The third set of one-dimensional tests is performed
for the constant-coe(cid:14)cient di(cid:11)usion equation
@ U =f(x); U(0) = ; U(1)= ; (3.11)
xx 0 1
which is de(cid:12)ned on the interval x 2 [0; 1], with the exact solution U = sin(x),
f =(cid:0)sin(x). FVD equations are formed at the interior dual volumes as
F (cid:0)F =cos(s )(cid:0)cos(s ); i=1; :::;N: (3.12)
si+1 si i+1 i
Fluxes approximatingthe solution derivative are de(cid:12)ned as
Ui(cid:0)Ui(cid:0)1
F = ; i=1; :::;N; (3.13)
si xi(cid:0)xi(cid:0)1
where U is a discrete approximation to U(x ) and the forcing term is integrated
i i
exactly. Dirichlet boundary conditions are strongly enforced as
U = ; U = : (3.14)
0 0 N+1 1
Primal Dual Grid-re(cid:12)nement computations
Mesh Mesh Discr. Error Trunc. Error
Uniform Unperturbed O(h2) O(h2)
Uniform Perturbed O(h2) O(1)
Random Unperturbed O(h2) O(h)
Random Perturbed O(h2) O(1)
Table 3.2
Convergence ofdiscretization and truncation errors for thedi(cid:11)usion equation.
Table 3.2 summarizes the discretization and truncation error convergence. The
mainobservationisthat the discretizationerrorsconvergewith secondorderevenfor
those irregular grids for which truncation errorsare O(1).
4. Two-dimensionalcomputations. Inthissection,wedemonstrate2Dcom-
putations for the Poisson equation and for a constant-coe(cid:14)cient advection equation
on general irregular grids. More details and other computations, including inviscid
(cid:13)ow equations on general irregular grids, can be found in [1, 8, 20].
4.1. Poisson equation. Awidelyusedsecond-orderFVDschemeisshownhere
for the Poisson equation,
(cid:1)U =f; (4.1)
with Dirichlet boundary conditions. In the FVD scheme, the conservation law
rU(cid:1)n^d(cid:0)= fd(cid:10) (4.2)
I ZZ
(cid:0) (cid:10)
is enforced on node-centered control volumes constructed by the median-dual parti-
tion; n^ is the outwardunit normal to the control-volumesurface. In distinction from
the thin-layer approximation considered in [18], the scheme uses the Green-Gauss
9
P
1
P
2
P
PP C
00
n
P P L
A B
n P
P R 4
3
Fig. 4.1. Illustration ofgradient reconstruction for viscoustermsonmixed gridswithmedian-
dual partition.
approachto provide accurate approximationsto the normal gradientsat the control-
volumeboundaries. Aseriesofre(cid:12)nedmixed-elementgridscomposedoftrianglesand
quadrangles is considered.
With reference to Figure 4.1, the integral(cid:13)ux throughthe dual faces adjacent to
the edge [P ;P ] is computed as
0 4
rU(cid:1)n^d(cid:0)=rUR(cid:1)nR+rUL(cid:1)nL; (4.3)
Z
ABC
where nR and nL are directed areasof the correspondingdual faces. The gradient is
reconstructed separately at each dual face as follows.
Forthetriangularelementcontribution,thegradientisdeterminedfromaGreen-
Gauss evaluation at the primal-grid element,
rUL =rU : (4.4)
014
The gradient overbar denotes a gradient evaluated by the Green-Gauss formula on
the primal cell identi(cid:12)ed by the point subscripts. With fully-triangular elements,
the formulation is equivalent to a Galerkin (cid:12)nite element scheme with a linear basis
function [2, 4]. For the quadrilateral element contribution, the gradient is evaluated
as
U (cid:0)U
rUR =rU + 4 0 (cid:0)rU (cid:1)e e ; (4.5)
0234 (cid:20)jr (cid:0)r j 0234 04(cid:21) 04
4 0
where r is the coordinate vector of node P and
i i
r (cid:0)r
e = 4 0 (4.6)
04 jr (cid:0)r j
4 0
is the unit vector aligned with the edge [P ;P ]. Note that for grids with dual faces
0 4
perpendicular to the edges, the edge-gradientis the only contributor. This approach
to the gradient reconstruction is used to decrease the scheme susceptibility to odd-
even decoupling [11, 12]. In all cases, the linear solution reconstruction leads to a
(cid:12)rst-order (cid:13)ux (gradient) reconstruction accuracy.
10