Realizability moments

In general, simulations carried out starting from realizable moment sets should result in realizable moment sets. Nevertheless, the moment-transport equations are integrated numerically with some finite discretization errors. As Wright (2007) clearly reports, most of the problems are caused by the approximation of the convective term, in particular with higher-order discretization schemes, which can turn a realizable set of moments into... 

As discussed above for a Gaussian distribution, this system of equations can be solved iteratively to determine o- and for a = 1,2 given a realizable moment set mk,... 

There are undoubtedly other examples for which the DQMOM fails but the QMOM works. However, as a general rule, the more dominant the diffusion component of the spatial fluxes, the more likely it is that the abscissas will exhibit smooth variations in space and the DQMOM will be applicable. In cases for which the DQMOM is applicable, the problem of realizable moment sets is fairly simple to address because the only requirement is that the weights be nonnegative. Nevertheless, it is still necessary in multivariate cases to verify that the abscissas are physically realizable, i.e. they must take on values in the phase space of the internal coordinates (see Section 8.1.1). With the DQMOM (as with the QMOM), there is no guarantee that the solutions will represent realizable quadratures for a multivariate NDF. 

In general, the volume-average transported moment set is related to a volume-average NDF. Thus, in order to show that is a realizable moment set, it suffices to show that is nonnegative. This idea of using the NDF to show the realizability of the moment set is the basis of kinetics-based FVM. 

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