Multivariate quadrature

But mi is usually not zero when the internal coordinate represents particle mass, surface area, size, etc. In these cases the PD algorithm can be safely used. The case of null mi occurs more often when the internal coordinate is a particle velocity that, ranging from negative to positive real values, can result in distributions with zero mean velocity. Another frequent case in which the mean is null is when central moments (moments translated with respect to the mean of the distribution) are used to build the quadrature approximation. These cases will be discussed later on, when describing the algorithms for building multivariate quadratures. 

These concepts can be used to construct a multivariate quadrature. 

Figure 3.3. The first step in the construction of the multivariate quadrature in the CQMOM. Determination of the N nodes and N weights for the first internal coordinate...

The construction of the multivariate quadrature begins with the calculation of the univariate quadrature of order N for the first internal coordinate by using the Wheeler algorithm with the first 2N - 1 moments ... 

Quadrature-based moment methods (QBMM) constitute a particular class of very successful moment methods that overcome the closure problem by using quadrature approximations. A general (multivariate) quadrature of N nodes for M internal coordinates requires knowledge of N weights, w , and N nodes (or abscissas), y M,a),... 

If the realizability condition in Eq. (8.52) is satisfied, found from Eq. (8.53) is guaranteed to be realizable. Using the method described above for the PBE (Eq. (8.35)), it is straightforward to extend Eq. (8.53) to second-order time-stepping. The extension to multiple velocity components v in multiple spatial dimensions is a bit more complicated. As described in Yuan Fox (2011) and in Section B.3 of Appendix B, when the CQMOM is used to constmct the multivariate quadratures all permutations must be used in a consistent manner in order to get the correct kinetic energy fiuxes. Nevertheless, for each permutation of the CQMOM, the basic time-stepping formula in Eq. (8.53) is used to update the transported moment set by modifying the definition of K , to include the multivariate moments in the optimal-moment set. Readers interested in more details on multidimensional free transport should consult Yuan Eox (2011) and Section B.3 of Appendix B. 

If the CQMOM is used for the multivariate quadrature reconstruction, another important technical point to keep in mind is that Eq. (B.13) corresponds to one permutation of the CQMOM. In other words, each of the P permutations of the CQMOM will generate a different representation of Nevertheless, the formulas given below for... 

When the CQMOM is used for the multivariate quadrature, we must accommodate the two CQMOM permutations ... 

In this definition, n -ifiy, fi) is reconstructed using extended multivariate quadrature from the transported moment set Since nf-ifiw,fi) has a known functional form, the halfmoment sets in Eq. (B.54) can be evaluated analytically. Once these half-moment sets are known, the Af-node multivariate quadrature is applied to find the two set of weights and abscissas. The overall procedure is as follows ... 

Buffo, A., Vanni, M., Marchisio, D. L. Fox, R. O. 2013 Multivariate quadrature-based moments methods for turbulent polydisperse gas-liquid systems. International Journal of Multiphase Plow (accepted for publication). 

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