A bin integral method for solving the kinetic collection equation

Abstract

A new numerical method for solving the kinetic collection equation (KCE) is proposed, and its accuracy and convergence are investigated. The method, herein referred to as the bin integral method with Gauss quadrature (BIMGQ), makes use of two binwise moments, namely, the number and mass concentration in each bin. These two degrees of freedom define an extended linear representation of the number density distribution for each bin following Enukashvily (1980). Unlike previous moment-based methods in which the gain and loss integrals are evaluated for a target bin, the concept of source–bin pair interactions is used to transfer bin moments from source bins to target bins. Collection kernels are treated by bilinear interpolations. All binwise interaction integrals are then handled exactly by Gauss quadrature of various orders. In essence the method combines favorable features in previous spectral moment-based and bin-based pair-interaction (or flux) methods to greatly enhance the logic, consistency, and simplicity in the numerical method and its implementation. Quantitative measures are developed to rigorously examine the accuracy and convergence properties of BIMGQ for both the Golovin kernel and hydrodynamic kernels. It is shown that BIMGQ has a superior accuracy for the Golovin kernel and a monotonic convergence behavior for hydrodynamic kernels. Direct comparisons are also made with the method of Berry and Reinhardt (1974), the linear flux method of Bott (1998), and the linear discrete method of Simmel et al. (2002).