The generalized quadrature method of moments

Abstract

The quadrature method of moments (QMOM) for a one-dimensional (1-D) population balance equation was introduced by R. McGraw (Aerosol Science and Technology, 27, 255-265, 1997) to close the moment source terms. QMOM is defined based on the properties of the monic orthogonal polynomials Qi of degrees i=0,1,…,n that are uniquely defined by the set of 2n moments up to order 2n−1. The moment of order 2n is fixed to the boundary of moment space such that the distribution function is approximated by a sum of n Dirac delta functions. Using the recursion coefficients of the orthogonal polynomials for i>n≥1, the generalized quadrature method of moments (GQMOM) extends the quadrature representation to a sum of N>n terms using the same moments as QMOM. In doing so, the known moments are preserved and higher-order moments correspond to a distribution function in the interior of moment space. Here, GQMOM closures for distributions on R, R+, and (0,1) are defined and analyzed. Generally speaking, GQMOM provides a more accurate moment closure than QMOM without increasing the number of moments and at nearly the same computational cost.