Weak convergence analysis for non-linear collisional induced breakage equation with singular kernel

TL;DR

This paper analyzes the weak convergence of a finite volume scheme for a nonlinear collisional breakage equation with singular kernels, ensuring conservation and non-negativity of solutions.

Contribution

It introduces a finite volume scheme for the nonlinear CBE with singular kernels and proves its weak convergence and solution properties.

Findings

The scheme conserves mass and non-negativity.

Weak convergence of the scheme to the continuous solution is established.

A stable time step condition is identified for convergence.

Abstract

The phenomenon of collisional breakage in particulate processes has garnered significant interest due to its wide-ranging applications in fields such as milling, astrophysics, and disk formation. This study investigates the analysis of the pure collisional breakage equation (CBE), characterized by its nonlinear nature with presence of locally bounded collision kernels and singular breakage kernels. Employing a finite volume scheme (FVS), we discretize the continuous equation and investigate the weak convergence of the approximated solution of the conservative scheme towards the continuous solution of CBE. A weight function is introduced to ensure the conservation of the scheme. The non-negativity of the approximated solutions is also shown with the assistance of the mathematical induction approach. Our approach relies on the weak $L^1$ compactness argument, complemented by introducing a stable condition on the time step.