A Finite Volume Scheme for the Solution of a Mixed Discrete-Continuous Fragmentation Model

Abstract

This paper concerns the construction and analysis of a numerical scheme for a mixed discrete-continuous fragmentation equation. A finite volume scheme is developed, based on a conservative formulation of a truncated version of the equations. The approximate solutions provided by this scheme are first shown to display conservation of mass and preservation of nonnegativity. Then, by utilising a Dunford--Pettis style argument, the sequence of approximate solutions generated is shown, under given restrictions on the model and the mesh, to converge (weakly) in an appropriate $L_1$ space to a weak solution to the problem. Additionally, by applying the methods and theory of operator semigroups, we are further able to show that weak solutions to the problem are unique and necessarily classical (differentiable) solutions. Finally, numerical simulations are performed to investigate the performance of the scheme and assess its rate of convergence.