Approximate solutions to the nonlinear hyperbolic population balance equation: convergence, error estimate and numerical simulations

We consider a nonlinear, hyperbolic population balance equation that incorporates both aggregation and collisional breakage events simultaneously. Our approach revolves around the development of a novel time-explicit finite volume scheme. Under a suitable time-step stability condition, we prove the convergence of the approximate solution for any non-uniform mesh. A first-order convergence is obtained by a thorough error analysis of the proposed scheme for a suitable choice of kernels. Finally, we compute some numerical test examples to explore the behavior of the solution in steady-state conditions as well as the occurrence of gelation phenomena.