The generalized quadrature method of moments. (January 2023) - Fox, Rodney O.; Laurent, Frédérique; Passalacqua, Alberto
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 Q i of degrees i = 0, 1, …, n that are uniquely defined by the set of 2 n moments up to order 2 n − 1 . The moment of order 2 n 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. Highlights: GQMOM improves the performance and accuracy of QMOM by allowing for an arbitrary number of quadrature nodes. GQMOM requires only one additional moment compared to QMOM with nearly the same computational cost. GQMOM uses the same moments as EQMOM but is more accurate and easier to implement. GQMOM can solve population balance equations for which QMOM and EQMOM have difficulties.