Multivariate quadrature CQMOM

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...

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 ... 

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