Finite volume convergence analysis and error estimation for non-linear collisional induced breakage equation

TL;DR

This paper analyzes the finite volume method's convergence and error estimation for a non-linear collisional breakage equation in particulate processes, providing theoretical and numerical validation of the scheme's accuracy.

Contribution

It introduces a weak convergence analysis and explicit error estimates for the FVM applied to the non-linear collisional breakage equation with non-conservative approximation.

Findings

Proves weak convergence under stability conditions

Provides explicit error bounds for the scheme

Numerical examples confirm first-order convergence

Abstract

This article focuses on the finite volume method (FVM) as an instrument tool to deal with the non-linear collisional-induced breakage equation (CBE) that arises in the particulate process. Notably, we consider the non-conservative approximation of the CBE. The analysis of weak convergence of the approximated solutions under a feasible stability condition on the time step is investigated for locally bounded breakage and collision kernels. Subsequently, explicit error estimation of the FVM solutions in uniform mesh having the kernels in the class of $W_{loc}^{1,\infty}$ space. It is also shown numerically for the first-order convergent scheme by taking numerical examples.