Convergence and error analysis for pure collisional breakage equation

TL;DR

This paper analyzes the convergence and error of a finite volume scheme for the nonlinear pure collisional breakage equation, providing theoretical proofs and numerical verification of first-order accuracy.

Contribution

It develops a weak convergence analysis for non-uniform meshes and establishes first-order error estimates for uniform meshes with specific kernel regularity.

Findings

Weak $L^1$ compactness ensures convergence of the scheme.

The scheme achieves first-order convergence rate.

Numerical tests confirm theoretical error estimates.

Abstract

Collisional breakage in the particulate process has a lot of recent curiosity. We study the pure collisional breakage equation which is nonlinear in nature accompanied by locally bounded breakage kernel and collision kernel. The continuous equation is discretized using a finite volume scheme (FVS) and the weak convergence of the approximated solution towards the exact solution is analyzed for non-uniform mesh. The idea of the analysis is based on the weak $L^1$ compactness and a suitable stable condition on time step is introduced. Furthermore, theoretical error analysis is developed for a uniform mesh when kernels are taken in $W_{loc}^{1,\infty}$ space. The scheme is shown to be first-order convergent which is verified numerically for three test examples of the kernels.