## A predictor–corrector algorithm for the coupling of stiff ODEs to a particle population balance

In this paper a novel predictor–corrector algorithm is presented for the solution of coupled gas-phase – particulate systems. The emphasis of this work is the study of soot formation, but the concepts can be applied to other systems. This algorithm couples a stiff ODE solver to a Monte Carlo population balance solver. Such coupling has been achieved previously for similar systems using a Strang operator splitting algorithm, however, that algorithm demonstrated several numerical issues which resulted in a high computational cost to acquire adequate precision. In particular a source-sink instability was identified whereby a largemagnitude source term present in the ODE system was competing with a similarly sized sink term in the population balance. This instability required that the splitting step size was very small in order to keep numerical error sufficiently low. A predictor–corrector algorithm has been formulated to negate this instability. An additional efficiency is gained with this algorithm as a principal computational cost of the Strang splitting algorithm is removed: the requirement to re-initialise the ODE solver every splitting step. The numerical convergence of the new algorithm is demonstrated, and its efficiency is compared to that of the Strang splitting algorithm. Substantial computation time savings are demonstrated, which allow a fixed error in three studied system functionals to be achieved with an order-of-magnitude reduction in computation time.

- This paper draws from the preprint: A predictor-corrector algorithm for the coupling of stiff ODEs to a particle population balance.

Keywords: Mathematical Modelling and Simulation, particle formation, population balance,

Associated Projects: Numerics, Nanoparticles, and Particle Processes