Moment-inversion algorithm DQMOM

In this expression, the weights and abscissas are found from the moments at every time step and at every spatial location using a moment-inversion algorithm. Following the method described in Marchisio Fox (2005), the DQMOM equations corresponding to Eq. (3.143) are... 

The quadrature method of moments (QMOM) and the direct quadrature method of moments (DQMOM) were introduced in Chapter 3 as equivalent methods for solving a homogeneous GPBE. In fact, the DQMOM was derived by Marchisio Fox (2005) primarily for the purpose of solving spatially inhomogeneous multivariate moment-transport equations. Unlike for the univariate case, where the moment-inversion algorithm is uniquely defined for a given set of moments, the QMOM in the multivariate case is much... 

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. 

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). 

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