Method of moments EQMOM

The value of cr is determined by fixing one additional moment (a total of 21V + 1 moments, i.e. an odd number of moments). In order to distinguish between moment methods using Eq. (3.81) and those using Eq. (3.82), we will refer to the former as the quadrature moment of moments (QMOM) and the latter as the extended quadrature method of moments (EQMOM) (Yuan et al, 2012). The principal advantage of using the EQMOM instead of the QMOM is that with one additional moment it is possible to reconstruct a smooth, nonnegative NDF that exactly reproduces the first 21V + 1 moments. However, there are several... 

Chapter 3 provides an introduction to Gaussian quadrature and the moment-inversion algorithms used in quadrature-based moment methods (QBMM). In this chapter, the product-difference (PD) and Wheeler algorithms employed for the classical univariate quadrature method of moments (QMOM) are discussed, together with the brute-force, tensor-product, and conditional QMOM developed for multivariate problems. The chapter concludes with a discussion of the extended quadrature method of moments (EQMOM) and the direct quadrature method of moments (DQMOM). 

The application of QBMM to Eq. (C.l) will require a closure when m(7 depends on 7 Nevertheless, the resulting moment equations (used for the QMOM or the EQMOM) and transport equations for the weights and abscissas (used for the DQMOM) will still be hyperbolic. In terms of hyperbolic conservation laws, the moments are conserved variables (which result from a linear operation on /), while the weights and abscissas are primitive variables. Because conservation of moments is important to the stability of the moment-inversion algorithms, it is imperative that the numerical algorithm guarantee conservation. For hyperbolic systems, this is most easily accomplished using finite-volume methods (FVM) (or, more specifically, realizable FVM). The other important consideration is the accuracy of the moment closure used to close the function, as will be described below. 

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