Ansari, ZeeshanZeeshanAnsariRae, MitchellMitchellRaeSriwastav, NikhilNikhilSriwastavDewangan, MainendraMainendraDewanganTomar, SaurabhSaurabhTomarHeinrich, StefanStefanHeinrichRamachandran, RohitRohitRamachandranWalker, GavinGavinWalkerSingh, MehakpreetMehakpreetSingh2026-05-122026-05-122026-03-10Computational Mechanics (in Press): (2026)https://hdl.handle.net/11420/63015Population balance equations (PBEs) have previously been applied for modelling across different fields including astrophysics for cloud formation, bubble dynamics, and liquid-liquid dispersion processes. In this paper, a generalised numerical method is presented by enabling the efficient solution of aggregation, breakage, growth, and nucleation mechanisms. A volume-conserving finite volume approximation is developed for the aggregation and breakage processes while the method of characteristics is used for the growth process. This addresses the instability in solutions of combined breakage-growth or aggregation-growth PBEs. Additionally, a nuclei-size cell is introduced at each time step to handle the combined aggregation-nucleation and breakage-nucleation processes. This approach eliminates the need to convert the original aggregation and breakage population balances into divergence form, allowing stable numerical solutions to be obtained for combined processes [Math. Models Methods Appl. Sci. 23(7), 235–1273]. The new numerical approximations are robust for handling complex simultaneous processes, effectively reducing numerical diffusion and dispersion. The accuracy and efficiency of the proposed scheme are validated through comparisons with newly derived analytical results and existing finite volume schemes, fixed pivot technique, and cell average technique, demonstrating its superiority in performance. The comparison shows that the new schemes predict number density functions (NDFs) and their integral moments with significantly higher precision. Computationally, the proposed approach captured all numerical results by consuming 40–70% lesser computational time compared to existing schemes.en1432-09242026Springerhttps://creativecommons.org/licenses/by/4.0/Cell average techniqueFinite volume schemesFixed pivot techniqueNonlinear integro-partial differential equationsPopulation balancesNatural Sciences and Mathematics::519: Applied Mathematics, ProbabilitiesNatural Sciences and Mathematics::530: PhysicsTechnology::660: Chemistry; Chemical EngineeringJournal Articlehttps://doi.org/10.15480/882.1708810.1007/s00466-026-02762-110.15480/882.17088