Thompson, Warsi, Mastin - Numerical Grid Generation, страница 15

In fact, as it is shown later in this chapter and in chapterV, the metric quantities should be represented numerically even when analytical expressionsare available. One might have, for example,(24)3. Special PointsMany of the expressions given in the previous section break down at so-called "specialpoints" in the field where special attention is required in the approximation of derivatives.These points commonly arise when geometrically complicated physical domains areinvolved. As indicated in Chapter II, special points can occur on the domain boundary andon interfaces between subregions of a composite curvilinear coordinate system.

They may berecognized in physical space as those interior points having a nonstandard number ofimmediate neighbors or, equivalently, those points which are vertices, or the center, of a cellwith either a nonstandard number of faces or a vertex shared by a nonstandard number ofother cells. (In two dimensional domains, ordinary interior points have eight immediateneighbors [refer to figure on p.141]; standard two-dimensional interior grid cells have foursides and share each vertex with three other cells [see diagram on p. 143].) Boundary pointsare not special unless they are vertex-centered and have a nonstandard number of immediateneighbors (other than five in two dimensions see diagram on p.

142 for an ordinary boundarypoint) and then are special only when their assocciated boundaryconditions contain spatialderivatives. Some examples of special cell-centered points and special vertex-centered pointsare shown below.When a finite-difference formulation is used, the usual approach, as described inSection 2, can be followed at a special point P if the transformed equations and differenceapproximations at that point are rephrased in terms of suitable local coordinates. The localsystem is chosen so as to orient and label only the surrounding points to be used in theneeded difference expressions.

Choices appropriate to various special points are listed inTables 1, 2, and 3.The difficulties encountered at special points in a finite-volume approach are clearlyseen by considering the image in the transformed plane. The first pair of diagrams below, forexample, shows that at centers of cells having the usual number of faces but sharing a vertexwith a nonstandard number of cells, such difficulties amount to mere bookkeepingcomplications when only first partials must be approximated.

Equations (17) and (18) stillapply, but the indices must be defined to correctly relate the cell centers on the two sides ofan interface. The following diagramsalso illustrate the breakdown at all special cell-centered points of the previously-describedfinite-volume schemes for approximating second and mixed partial derivatives. This isbecause the forward/backward orientation of the coordinate system in one segment cannot beconsistently followed across the interface adjacent to, or intersecting, the special points.

Thesecond pair of diagrams displays the additional complication associated with grid cellshaving a nonstandard number of edges. Such a cell can occur on an interface betweensegments of a composite grid which are joined between grid lines. When the segments aretransformed to their respective images, the separate pieces of the special grid cell cannot bejoined without distorting them. It is thus unclear how to evaluate the volume and theoutward normals of that transformed cell in order to use identity (16) in the transformedplane. Consequently, at special points of this type and at all special points where secondderivatives must be approximated, the governing equations are best represented locally in thephysical plane where such ambiguities do not exist.Treatment in physical space involves approximation of the original equations bymeans of identity (16).

Thus, for a two-dimensional N-sided cell of area A with cartesiancentroid P = (p1,p2), verticesi=1,2,...,N, and edges si joining Vi and Vi+1(VN+1=V1) along which a function f and its first partial derivatives are constant, thisapproach giveswhere the superscripts on f and its derivatives indicate the point or face of evaluation. As inithe previous section, an obvious way to approximate f s is to average the center values of thetwo cells sharing edge si.

This same averaging scheme cannot be repeated to approximate, and, however, without rejecting the recommended strategy of avoiding use ofvalues at points which are not immediate neighbors of the point at which a quantity is beingevaluated. Instead, we propose the averaging technique:where the vertex values are obtained by applying identity (16) to auxiliary cells formed byjoining the midpoints of the edges of each cell to the cell center. To make this more precise,let V be a vertex common to Q cells and label the cell faces emanating from V as ki withmidpointsThen ifis the center of the cell having edges ki and ki+1, and ifthe first partial derivatives of f at V may be approximated bywhere A is the area of the 2Q-faced auxiliary cell M1P1M2P2...MQPQM1 indicated in thefollowing diagram.This technique is applicable to all grid cell centers; however, it is recommended for use onlyat points where the methods developed in section 2 break down, since the differencerepresentations associated with those methods are simpler.4.

Metric IdentitiesWhen the transformed equations are in conservative form, it is possible for the metriccoefficients to introduce spurious source terms into the equations, as has been noted inseveral works cited in Ref. [1] and as discussed also in Ref.

[11] and [12]. This is becausethe metric coefficients have been included in the operand of the differential operators and ifthe differencing of these coefficients does not numerically satisfy identities (III-40) and(III-120), the numerical representations of derivatives of uniform physical quantities arenonvanishing.For example, if the quantity A is constant, the conservative form for the gradient, Eq.(III-42) giveswhich is precisely Eq.

(III-40). Relations (III-43) - (III-45) similarly reduce to (III-40) whenis uniform. Therefore, Eq. (III-40), or equivalently Eq. (III-21), is a metric identity whichmust be satisfied numerically in order that the conservative expressions for the gradient,divergence, curl, and Laplacian, etc., vanish when the physical variable is uniform.

Thisconsideration does not arise with the non-conservative forms since the quantity A isdifferentiated directly in those expressions.Another metric identity which must be satisfied numerically arises when the grid istime-dependent. This may be seen by considering a generic conservation equation of theformThe conservative relation (III-121) transforms this to(25)where now the time derivative is understood to be at a fixed point in the transformed space.If A and are both constants, then Eq. (25) giveswhich vanishes according to Eq.

(III-40). Expansion of the left-hand summation subject toEq. (III-40) then reveals the additional identity to be satisfied:(26)which is just Eq. (III-120). This equation, therefore, is that which should be used towhich is just Eq. (III-120). This equation, therefore, is that which should be used tonumerically determine updated values of the Jacobian, . For ifis instead updateddirectly from the new values of the cartesian coordinates, spurious source terms will appear.The following example provides a simple illustration of differencing schemes whichdo, and do not, satisfy the metric identities.

The conservative expression for a first derivativein two dimensions is given in Eq. (III-96) as(27)which for uniform f reduces to(28)Suppose that f x is to be represented at the center of the cell shown below.The differencing scheme should satisfyOne possible candidate is the sequence of central differences represented by(29a)(29b)The resulting expressions for the mixed partials arewhich are indeed equal and thus satisfy identity (28).

An alternate choice might be to usecentral differences for the second differentiation as in Eq. (29a), while approximating therequired edge values of the first partials by the average of the values at the adjacent nodes,e.g.The nodal values are reasonably represented by central differences such asThis scheme cannot possibly satisfy (28), however, since the points used to representare:while those needed to evaluateare:It should be noted that the representations in both of these schemes are consistent andof the same formal order of accuracy. Also, if the metric coefficients at the grid points wereevaluated and stored, it would perhaps be natural to follow the second approach, usingaverages of the metric coefficients at the intermediate points.

This, however, is notacceptable since it fails to satisfy the metric identity involved and thus would introducespurious non-zero gradients in a uniform field.This example suggests one basic rule that should always be followed: Never averagethe metric coefficients. Rather, average the coordinate values themselves, if necessary, andthen calculate the metric derivatives directly.

Alternatively, a coordinate system can begenerated with mesh points at all of the half-integer points, as well as at the integer pointsused in the physical solution. The metric coefficients can then be evaluated directly bydifferencing between neighboring points, even at the half-integer points. For example,This approach was used in Ref. [13] and problems with the metric identities were therebyeliminated.It is also possible to construct difference representations which do not involve anyaveraging and yet still do not satisfy the metric identities; schemes which use unsym metricdifferences are an example.

Fortunately, most reasonable symmetric expressions withoutaveraging do satisfy the identities.In the representation of the Laplacian using Eq. (III-71), 2 i should be calculatedusing Eq. (III-74), rather than using derivatives of the metric tensor elements.Caution is required even when the coordinate transformation is known explicitly. Inthat case, the metric coefficients can be evaluated analytically, but the metric identities willnot in general be satisfied numerically when these coefficients are differenced. This is trueeven in the simple case of cylindrical coordinates as the following example shows. Withthe partials of y( , ) areIf the first partial derivative f x is represented as in Eq.

(27) and the difference in Eq. (29a) isused, but the first partials y and y are represented exactly, e.g.the bracket in Eq. (27) evaluated atfor uniform f becomeswhich does not vanish identically. Thus the metric identity (28) is not satisfied when themetric derivatives, y and y , are evaluated analytically. But it was shown above that thedifference form used here, Eq. (29a), does in fact satisfy the metric identity (28) when themetric derivatives are evaluated numerically without averaging.The use of exact analytical expressions for the metric coefficients therefore does notnecessarily increase the accuracy of the difference representations, and may actually degradethe accuracy.