A Flux Method for the Numerical Solution of the Stochastic Collection Equation

Abstract
A new mass conservative flux method is presented for the numerical solution of the stochastic collection equation. The method consists of a two-step procedure. In the first step the mass distribution of drops with mass x′ that have been newly formed in a collision process is entirely added to grid box k of the numerical grid mesh with xk ⩽ x′ ⩽ xk+1. In the second step a certain fraction of the water mass in grid box k is transported to k + 1. This transport is done by means of an advection procedure. Different numerical test runs are presented in which the proposed method is compared with the Berry–Reinhardt scheme. These tests show a very good agreement between the two approaches. In various sensitivity studies it is demonstrated that the flux method remains numerically stable for different choices of the grid mesh and the integration time step. Since a time step of 10 s may be used without significant loss of accuracy, the flux method is numerically very efficient in comparison to the Berry–Re... Abstract A new mass conservative flux method is presented for the numerical solution of the stochastic collection equation. The method consists of a two-step procedure. In the first step the mass distribution of drops with mass x′ that have been newly formed in a collision process is entirely added to grid box k of the numerical grid mesh with xk ⩽ x′ ⩽ xk+1. In the second step a certain fraction of the water mass in grid box k is transported to k + 1. This transport is done by means of an advection procedure. Different numerical test runs are presented in which the proposed method is compared with the Berry–Reinhardt scheme. These tests show a very good agreement between the two approaches. In various sensitivity studies it is demonstrated that the flux method remains numerically stable for different choices of the grid mesh and the integration time step. Since a time step of 10 s may be used without significant loss of accuracy, the flux method is numerically very efficient in comparison to the Berry–Re...