All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 446 175 7
PDF Downloads 312 116 6

An Efficient Numerical Solution to the Stochastic Collection Equation

Shalva Tzivion (Tzitzvashvili)Department of Geophysics and Planetary Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, 69978, Israel

Search for other papers by Shalva Tzivion (Tzitzvashvili) in
Current site
Google Scholar
PubMed
Close
,
Graham FeingoldDepartment of Geophysics and Planetary Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, 69978, Israel

Search for other papers by Graham Feingold in
Current site
Google Scholar
PubMed
Close
, and
Zev LevinDepartment of Geophysics and Planetary Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, 69978, Israel

Search for other papers by Zev Levin in
Current site
Google Scholar
PubMed
Close
Full access

Abstract

A new, accurate, efficient method for solving the stochastic collection equation (SCE) is proposed. The SCE is converted to a set of moment equations in categories using a new analytical form of Bleck&'s approach. The equations are written in a form amenable to solution and to a category-by-category analysis of drop formation and removal. This method is unique in that closure of the equations is achieved using an expression relating high-order moments to any two lower order moments, thereby restricting the need for approximation of the category distribution function only to integrals over incomplete categories. Moments in categories are then expressed in terms of complete moments with the aid of linear or cubic polynomials. The method is checked for the case of the constant kernel and a linear polynomial kernel. Results show that excellent approximation to the analytical solutions for these kernels are obtained. This is achieved without the use of weighting functions and with modest computing time requirements. The method conserves two or more moments of the spectrum (as required) and successfully alleviates the artificial enhancement of the collection process which is a feature of many schemes.

Abstract

A new, accurate, efficient method for solving the stochastic collection equation (SCE) is proposed. The SCE is converted to a set of moment equations in categories using a new analytical form of Bleck&'s approach. The equations are written in a form amenable to solution and to a category-by-category analysis of drop formation and removal. This method is unique in that closure of the equations is achieved using an expression relating high-order moments to any two lower order moments, thereby restricting the need for approximation of the category distribution function only to integrals over incomplete categories. Moments in categories are then expressed in terms of complete moments with the aid of linear or cubic polynomials. The method is checked for the case of the constant kernel and a linear polynomial kernel. Results show that excellent approximation to the analytical solutions for these kernels are obtained. This is achieved without the use of weighting functions and with modest computing time requirements. The method conserves two or more moments of the spectrum (as required) and successfully alleviates the artificial enhancement of the collection process which is a feature of many schemes.

Save