Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 17
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This is a mootpoint with uniform spacing, but two senses of order on a nonuniform grid emerge: thebehavior of the error (1) as the number of points in the field is increased while maintainingthe same relative point distribution over the field, and (2) as the relative point distribution ischanged so as to reduce the spacing locally with a fixed number of points in the field.On curvilinear coordinate systems the definition of order of a difference representationis integrally tied to point distribution functions.
The order is determined by the errorbehavior as the spacing varies with the points fixed in a certain distribution, either byincreasing the number of points or by changing a parameter in the distribution, not simply byconsideration of the points used in the difference expression as being unrelated to each other.Actually, global order is meaningful only in the first sense, since as the spacing is reducedlocally with a fixed number of points in the field, the spacing somewhere else must certainlyincrease.
This second sense of order on a nonuniform grid then is relevant only locally inregions where the spacing does in fact decrease as the point distribution is changed.In the following sections an illustrative error analysis is given. The generaldevelopment from which this is taken appears in Ref. [17], together with references torelated work.1. Order On Nonuniform SpacingA general one-dimensional point distribution function can be written in the form(1)In the following analysis, x will be considered to vary from 0 to l. (Any other range of x canbe constructed simply by multiplying the distribution functions given here by an appropriateconstant.) With this form for the distribution function, the effect of increasing the number ofpoints in a discretization of the field can be seen explicitly by defining the values of at thepoints to be successive integers from 0 to N. In this form, N+1 is then the number of pointsin the discretization, so that the dependence of the error expressions on the number of pointsin the field will be displayed explicitly by N.
This form removes the confusion that can arisein interpretation of analyses based on a fixed interval, where variation of thenumber of points is represented by variation of the interval. The form of the distributionfunction, i.e., the relative concentration of points in certain areas while the total number ofpoints in the field is fixed, is varied by changing parameters in the function.Considering the first derivative in one dimension:(2)with a central difference for fnoted above):we have the following difference expression (with=1 as(3)where T1 is the truncation error. A Taylor series expansion then yields(4)Here the metric coefficient, x is considered to be evaluated analytically, and hence has noerror. (The case of numerical evaluation of the metric coefficients is considered in a latersection.)The series in (4) cannot be truncated without further consideration since the-derivatives of f are dependent on the point distribution.
Thus if the point distribution ischanged, either through the addition of more points or through a change in the form of thedistribution function, these derivatives will change. Since the terms of the series do notcontain a power of some quantity less than unity, there is no indication that the successiveterms become progressively smaller.It is thus not meaningful to give the truncation error in terms of -derivatives of f.Rather, it is necessary to transform these -derivatives to x-derivatives, which, of course, arenot dependent on the point distribution. The first -derivative follows from (2):(5)Then(6)and(7)Each term in fcontains three -differentiations.
This holds true for all higherderivatives also, so that each term in fwill contain five -differentiations, etc.A. Order with fixed distribution functionFrom Eq. (1) we have(8)Therefore if the number of points in the grid is increased while keeping the same relativewill be proportional to 1/N3 and eachpoint distribution, it is clear that each term in fterm in fwill be proportional to 1/N5, etc.It then follows that the series in Eq. (4) can be truncated in this case, so that thetruncation error is given by the first term, which is, using Eq. (6),(9)The first two terms arise from the nonuniform spacing, while the last term is the familiarterm that occurs with uniform spacing as well.From (9) it is clear that the difference representation (3) is second-order regardless ofthe form of the point distribution function, in the sense that the truncation error goes to zeroas 1/N2 as the number of points increases.
This means that the error will be quartered whenthe number of points is doubled in the same distribution function. Thus all differencerepresentations maintain their order on a nonuniform grid with any distribution of points inthe formal sense of the truncation error decreasing as the number of points is increased whilemaintaining the same relative point distribution over the field.The critical point here is that the same relative point distribution, i.e., the samedistribution function, is used as the number of points in the field is increased. If this is thecase, then the error will be decreased by a factor that is a power of the inverse of the numberof points in the field as this number is increased. Random addition of points will, however,not maintain order.
In a practical vein this means that with twice as many points the solutionwill exhibit one-fourth of the error (for second-order representations in the transformedplane) when the same point distribution function is used. However, if the number of points isdoubled without maintaining the same relative distribution, the error reduction may not be asgreat as one-fourth.From the standpoint of formal order in this sense there is no need for concern over theform of the point distribution. However, formal order in this sense relates only to thebehavior of the truncation error as the number of points is increased, and the coefficients inthe series may become large as the parameters in the distribution are altered to reduce thelocal spacing with a given number of points in the field.
Thus, although the error will bereduced by the same order for all point distributions as the number of points is increased,certain distributions will have smaller error than others with a given number of points in thefield, since the coefficients in the series, while independent of the number of points, aredependenton the distribution function.B. Order with fixed number of pointsAn alternate sense of order for point distributions is based on expansion of thetruncation error in a series in ascending powers of the spacing, x , with the number ofpoints in the grid kept fixed and the point distribution changed to decrease the local spacing.From Eq. (9) second-order requires that(10)This is a severe restriction that is unlikely to be satisfied.
This is understandable, however,since with a fixed number of points the spacing must necessarily increase somewhere whenthe local spacing is decreased.The difference between these two approaches to order should be kept clear. The firstapproach concerns the behavior of the truncation error as the number of points in the fieldincreases with a fixed relative distribution of points. The series there is a power series in theinverse of the number of points in the field, and formal order is maintained for all pointdistributions. The coefficients in the series may, however, become large for somedistribution functions as the local spacing decreases for any given number of points.
Theother approach concerns the behavior of the error as the local spacing decreases with a fixednumber of points in the field. This second sense of order is thus more stringent, but theconditions seem to be unattainable.2. Effect of Numerical Metric CoefficientsThe above analysis has assumed the use of exact values of x the metric coefficient.If the metric coefficient is evaluated numerically, we have, in place of Eq.
(3), the differenceexpression(11)The Taylor expansion yieldsor(12)The coefficient of f xx here is the difference representation of xwhile that of f xxx reducesto a difference expression of . We thus have T2 given by the last two terms of T1 and thefirst term of T1 has been eliminated from the truncation error by evalutating the metriccoefficient numerically rather than analytically.Thus the use of numerical evaluation of the coordinate derivative, rather than exactanalytical evaluation, eliminates the f x term from the truncation error. Since this term is themost troublesome part of the error, being dependent on the derivative being represented, it isclear that numerical evaluation of the metric coefficients by the same differencerepresentation used for the function whose derivative is being represented is preferable overexact analytical evaluation.
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