Kernel density function EQMOM

As an example of univariate EQMOM, we will consider an NDE with (-co, -1-00) and define kernel density functions using Gaussian distributions (Chalons et al, 2010) ... 

Univariate EQMOM can be applied to an NDE defined on a finite interval (Yuan et at., 2012). Eor example, if the NDE is nonzero only on the interval [0,1], then we can define the kernel density functions using a beta distribution ... 

While EQMOM allows us to capture an additional moment, the use of kernel density functions can lead to a closure problem when evaluating integrals such as Eq. (3.9) ... 

In principle, the EQMOM introduced in Section 3.3.2 can be generalized to include multiple internal coordinates. However, depending on the assumed form of the kernel density functions, it may be necessary to use a multivariate nonlinear-equation solver to find the parameters (i.e. similar to the brute-force QMOM discussed in Section 3.2.1). An interesting alternative is to extend the CQMOM algorithm described in Section 3.2.3. Here we consider examples using both methods. 

Even when multivariate EQMOM is used with kernel density functions, a Dirac delta function (dualquadrature) representation is employed to close the terms in the GPBE. See Section 3.3.4 for more details. 

To conclude this section, we discuss three technical points that arise when applying the realizable high-order scheme for free transport. The first point is how to choose the value of N given that iV = is used for multivariate EQMOM. To answer this question, we first note that the highest-order transported moment in any one direction is of order 2n (which corresponds to 2n -t-1 pure moments in any one direction). The extra (even-order) moment is used in the EQMOM to determine the spread of the kernel density functions (see Section 3.3.2 for details). When free transport is applied to the moment of order 2n, the flux function involves the moment of order 2n -i- 1. Therefore, in order to exactly predict the flux function for free transport using the half-moment sets, we must have fV > (n-i-1). The obvious choice for N is thus fV = (n -i- 1). ... 

In proposed applications of mixed advection to gas-particle flows (Pandya Mashayek, 2003b Reeks, 1977, 1983, 1993 Zaichik, 1997, 1999 Zaichik et al, 2008), A is taken to be independent of v, and hence it behaves as an effective pressure. Therefore, in order for the time step computed from Eq. (B.52) to be nonzero, it will be necessary for A to be zero at spatial locations where either or is zero. Since these abscissas are computed from the half-moment sets and respectively, they can be zero only if the kernel density functions used in tlie EQMOM have zero width. (See Section 3.3.2... 

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