Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 16
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In Chapter V it is shown further that a detrimental contribution to thetruncation error can be removed by evaluating the metric coefficients numerically rather thananalytically. Accuracy in the representation of the metric coefficients thus has no inherentvalue in and of itself. Rather, it is the accuracy of the overall difference representation that isimportant.To summarize, the metric identities can often be numerically satisfied through carefulattention to the evaluation of the metric coefficients. These coefficients should be expressedas differences, not by analytical expressions.
They should be evaluated directly fromcoordinate values wherever they are needed and should never be averaged, since the use ofaveraged values will almost certainly result in failure to satisfy the metric identities.Intermediate coordinate values needed to construct differences which are compatible withthe metric identities can be obtained by averaging the coordinates at neighboring grid pointsor by using a grid with twice as many points in each direction as to be used in the actualsolution.The metric identities may become more difficult to satisfy numerically in threedimensions and in schemes involving higher-order operators or unsymmetric differenceexpressions, as may be needed at boundaries or near the special points discussed previously.When exact satisfaction is not achieved, the effects of the spurious source terms can bepartially corrected, as discussed in Ref.
[12], by subtracting off the product of the metricidentities with either a uniform solution or the local solution. The former amounts to using akind of perturbation form, while the latter is, in effect, expansion of the product derivativesinvolving the metric coefficients and retention of the supposedly vanishing terms, thusputting the equations into a weak conservation law form. Thus the gradient could be written,using Eq.
(III-42), aswhere AO is either the local value of A or a uniform value. In view of Eq. (III-40), thismodification does not change the analytical expression for the gradient. Differencerepresentations based on this form of the gradient will clearly vanish for uniform A equal toAO.
Analogous weakly-conservative expressions for all the other derivative operations ofChapter III can be inferred immediately. Subtraction of the product of the identity and theuniform free stream solution was used in Ref. [14], because of the difficulty in satisfying themetric identities exactly with flux-vector-splitting involving directional differences.All codes should be checked for the presence of spurious source terms arising from themetric identities by running with uniform non-zero values for all of the dependent variables.If such a test run produces any changes at all, some failure to satisfy the metric identities hasescaped detection (assuming the code is free from errors), and the difference representationsshould be modified or a change should be made to the weakly-conservative form describedabove.5.
Implementation ProcedureWhen a coordinate system has been generated, the values of the cartesian coordinates,xi will be available as functions of the curvilinear coordinates, i, with i = 1,2,3. Althoughthese relations might be in the form of analytical equations in the event that the coordinatesystem was generated by some analytical means, a more common result is a set of valuesgenerated by a numerical solution. By definition the curvilinear coordinates take on integervalues at the grid points ( i=0,1,2,...Ni where Ni+1 is the i total number of points in the idirection). Thus the values of x1,x2,x3 will be available at each grid point 1, 2, 3.Difference expressions, such as Eq.
(24), are then used at each grid point to evaluatethe components of the three covariant base vectors, from Eq: (III-3):As discussed previously, the metric derivatives should not be averaged, but rather shouldalways be evaluated directly from differences between grid points.
Therefore, it may benecessary in some difference formulations to have coordinate values available at pointsbetween the grid points on which the solution is to be represented. In that case, thecoordinate values at such points should be generated either by averaging the coordinatevalues between adjacent main points or by generating the coordinate system with twice asmany grid points in each direction as will be used in the solution representation.The nine elements of the covariant metric tensor can then be evaluated at each pointfrom Eq. (III-5):(Only six of these elements are distinct, of course, since the tensor is symmetric, so only sixdot products actually need to be evaluated.) The Jacobian is then evaluated at each pointusing Eq. (III-16):Next the three components of each of the three contravariant base vectors are evaluated ateach point from Eq. (III-33):and the nine elements of the contravariant metric tensor are evaluated at each point from Eq.(III-37):Again only six elements are distinct.All quantities involved in the transformed derivative operations are now available ati may be storedeach point.
Recall that if conservative forms are to be used, the productat each point, being evaluated fromto avoid the need for multiplication ofi byin all the operations.In transforming the physical partial differential equations, the gradient, divergence,curl, and Laplacian operations will have been replaced by either the conservativeexpressions, Eq. (III-42)-(III-45), or by the non-conservative expressions, Eq.(III-65)-(III-71).
Derivatives occurring individually will have been replaced by theexpressions given by Eq. (III-50)-(III-52). Finally, derivatives occurring in boundaryconditions will have been replaced by the expressions in Eq. (III-78), (III-79) or (III-82).Integrals will have been replaced by the relations given by Eq. (III-83)-(III-S7). Thus, withthe metric quantities evaluated at each point, as discussed above, all quantities involved inthe difference representations of the transformed partial differential equations are available.As was noted in Chapter III, the use of the conservative forms of the gradient,divergence, curl, Laplacian, etc. in the partial differential equations is equivalent toformulation of difference equations from the integral form of these equations.
Hencefinite-volume formulations may be set up directly from the partial differential equations byusing the conservative forms for the derivative operators involved.It should be pointed out again that the transformed partial differential equations are ofthe same form and type as the original equations, and are more complicated only in the senseof having variable coefficients, cross-derivatives, and more terms. The field on which theseequations are solved is rectangular and the grid is fixed, uniform and square. Therefore allnumerical solution algorithms that have been developed for partial differential equations oncartesian coordinate systems are applicable to these transformed equations, and all thesimplifications that result from the use of uniform square grids are in order, as well.Exercises1.
Verify Eq. (2).2. Apply Eq. (16) to the (i,j) cell in the diagram below this equation to obtain Eq. (17). In(16) interpret the gradient aswhereandare unit vectors in thetransformed space. The normal==1.of the cell. Recallisandordirections, respectively, in the, as appropriate, on each of the faces3. Following the procedure given with Eq. (20) and (21), obtain Eq.
(12) from Eq. (16)applied to the (i,j) cell. Show that the two mixed partials obtained in this manner are notequal, but that their average gives Eq. (15).4. Verify the boundary value, f i,j-1/2 given on p.1475. In cylindrical coordinates show that the conservative expression foruniform f when the metric coefficients are evaluated analytically.does not vanish forV. TRUNCATION ERRORDifference representations on curvilinear coordinate systems are constructed by firsttransforming derivatives with respect to cartesian coordinates into expressions involvingderivatives with respect to the curvilinear coordinates (the metric coefficients). Thederivatives with respect to the curvilinear coordinates are then replaced with differenceexpressions on the uniform grid in the transformed region.
The "order" of a differencerepresentation refers to the exponential rate of decrease of the truncation error with the pointspacing. On a uniform grid this concerns simply the behavior of the error as the pointspacing decreases. With a nonuniform point distribution, there is some ambiguity in theinterpretation of order, in that the spacing may be decreased locally either by increasing thenumber of points in the field or by changing the distribution of a fixed number of points.Both of these could, of course, be done simultaneously, or the points could even be movedrandomly, but to be meaningful the order of a difference representation must relate to theerror behavior as the point spacing is decreased according to some pattern.
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