Numerical analysis for coagulation-fragmentation equations with singular rates

TL;DR

This paper analyzes the convergence of a finite volume scheme for coagulation-fragmentation equations with singular rates, establishing weak solution convergence and error estimates under certain conditions, validated through numerical tests.

Contribution

It introduces a convergence proof for a finite volume scheme handling singular fragmentation rates and provides first-order error estimates when kernels are sufficiently smooth.

Findings

Convergence of the numerical scheme to a weak solution is proven.

A stable time step condition is identified for convergence.

Numerical experiments confirm theoretical results.

Abstract

This article deals with the convergence of finite volume scheme (FVS) for solving coagulation and multiple fragmentation equations having locally bounded coagulation kernel but singularity near the origin due to fragmentation rates. Thanks to the Dunford-Pettis and De La Vall$\acute{e}$e-Poussin theorems which allow us to have the convergence of numerically truncated solution towards a weak solution of the continuous model using a weak $L^1$ compactness argument. A suitable stable condition on time step is taken to achieve the result. Furthermore, when kernels are in $W^{1,\infty}_{loc}$ space, first order error approximation is demonstrated for a uniform mesh. It is numerically validated by attempting several test problems.