Central difference averaged

For stationary flows, the time-averaged values should be used in place of X, y in the central-difference formula in order to improve the smoothness of the estimated fields. For non-stationary flows, it may be necessary to filter out excess statistical noise in u Uj X, iy before applying (7.71). In either case, the estimated divergence fields are given by... 

Central differences Flux between control volumes is determined by the average condition of the two control volumes. 

To fully understand this concept of filtering, note that the values of flow properties at discrete points in a numerical simulation represent averaged values. To illustrate this explicitly, Rogallo and Moin [133] considered the central difference approximation for the first derivative of a continuous variable, v x), in a grid with points spaced a distance h apart. We can write this as follows (e.g., [133], p. 103 [186], p 323) ... 

This shows that the central-difference approximation can be thought of as an operator that filters out scales smaller than the mesh size. Furthermore, the approximation yields the derivative of an averaged value of v x). 

The the source terms are approximated by the midpoint rule, in which S is considered an average value representative for the whole grid cell volume. The derivatives are represented by an abbreviated Taylor series expansion, usually a central difference expansion of second order is employed. 

Stirling central differences represent a horizontal line using averages of the differences i.e.,... 

An alternative differencing scheme may be arrived at by considering the average derivative given by the forward and backward schemes this is called the central difference approximation ... 

The system is coupled to the central unit of a computer to ensure coordination of injection of the sample and collection of the data coming from the three detectors. Software has been developed to assist the analysis given from the detectors and to calculate the different average molecular weights and the polymolecularity index of the analysed species. Other interesting parameters... 

Equation 4 was discretised by a 5-point central difference formula and thereafter first-order differential equations 1, 2, 4 and 6 were solved by a backward difference method. Apparent reaction rate was solved by summing the average rates of each discretisation piece of equation 4. The reactor model was integrated in a FLOWBAT flowsheet simulator [12], which included a databank of thermodynamic properties as well as VLE calculation procedures and mathematical solvers. The parameter estimation was performed by minimising the sum of squares for errors in the mole fractions of naphthalene, tetralin and the sum of decalins. Octalins were excluded from the estimation because their content was low (<0.15 mol-%). Optimisation was done by the method of Levenberg-Marquard. 

The solution to this problem is available as Example 5.3 in the text. Table P5.7 shows the results obtained by the average acceleration and central difference methods. For this problem, both methods give the peak response with roughly the same accuracy. 

Comparing these chords with the actual tangent, it will be clear that the central difference formula, Eq. 3.12, is the most satisfactory on average (it is possible to draw three points where it it is not the best). Any of the above three formulae can be used in digital simulation. 

This method derives from the trapezium method in the ode field in which the time derivative in (8.9), expressed exactly as in (8.10), becomes a second-order central difference by virtue of the fact that the right-hand side now refers to a point in time midway in the time interval. This is achieved by taking the average of the second spatial derivative at the present time T and that at T - - 8T ... 

D = differential operator I = integral operator E = shift operator A = forward difference operator V = backward difference operator 6 = central difference operator p = averager operator. 

Application of the averager on the odd central differences gives he first averaged central difference as follows ... 

As expected, the effect of the averager is to remove the midpoint values of the function y from the odd central differences. 

The relationships between central difference operators and differential operators can now be developed, Eq. (3.73), representing the first averaged central difference, is combined with Eqs. (3.17) and (3.22) to yield... 

The higher-order averaged odd central difference operators are obtained by taking products of Eqs. (3.78) and (3.81). The higher-order even central differences are formulated by taking powers of Eq. (3.81). The third and fourth central operators, thus obtained, are listed below ... 

The odd-order differential operators in Eq. (3.127) are replaced by averaged central differences and the even-order differential operators by central differences, all taken from Table 3.3. Substituting these into Eq. (3.127) and regrouping of terms yield the formula... 

Other forms of Stirling s interpolation formula exist, which make use of base points spaced at half intervals (i.e., at h/2). Our choice of using averaged central differences to replace the odd differential operators eliminated the need for having base points located at the midpoints. The central differences for Eq. (3.129) are tabulated in Table 3.6. 

Express the differential and averaged central difference operators in terms of Iheir respective definitions ... 

In addition, the second-order partial derivative is expressed at the half point as a weighted average of the central differences at points (i, n+ 1) and (i, n) ... 

Fig. 1. The time evolution (top) and average cumulative difference (bottom) associated with the central dihedral angle of butane r (defined by the four carbon atoms), for trajectories differing initially in 10 , 10 , and 10 Angstoms of the Cartesian coordinates from a reference trajectory. The leap-frog/Verlet scheme at the timestep At = 1 fs is used in all cases, with an all-atom model comprised of bond-stretch, bond-angle, dihedral-angle, van der Waals, and electrostatic components, a.s specified by the AMBER force field within the INSIGHT/Discover program.

One further point might be made here. Although the example illustrates the difference between the two types of molecular weight average, the weight average molecular weight in this example cannot be said to be truly representative, an essential requirement of any measure of central tendency. In such circumstances where there is a bimodal, i.e. two-peaked, distribution additional data should be provided such as the modal values (100 and 100000 in this case) of the two peaks. 

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