2025/07/22 by Ziyao Xu, Xu, Ziyao, Guanyang Liu +3
Mathematics · Medicine · #Conservation law #Conservation of mass #Differential Equations and Numerical Methods #Discontinuous Galerkin method #Discretization #FOS: Mathematics #Galerkin method #Mathematical and Theoretical Epidemiology and Ecology Models #Moment (physics) #Numerical Analysis (math.NA) #Population #Population balance equation #Robustness (evolution)
paper · pdf · doi:10.48550/arxiv.2507.16631
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2025/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We develop a conservative, positivity-preserving discontinuous Galerkin (DG) method for the population balance equation (PBE), which models the distribution of particle numbers across particle sizes due to growth, nucleation, aggregation, and breakage. To ensure number conservation in growth and mass conservation in aggregation and breakage, we design a DG scheme that applies standard treatment for growth and nucleation, and introduces a novel discretization for aggregation and breakage. The birth and death terms are discretized in a symmetric double-integral form, evaluated using a common refinement of the integration domain and carefully selected quadrature rules. Beyond conservation, we focus on preserving the positivity of the number density in aggregation-breakage. Since local mass corresponds to the first moment, the classical Zhang-Shu limiter, which preserves the zeroth moment (cell average), is not directly applicable. We address this by proving the positivity of the first moment on each cell and constructing a moment-conserving limiter that enforces nonnegativity across the domain. To our knowledge, this is the first work to develop a positivity-preserving algorithm that conserves a prescribed moment. Numerical results verify the accuracy, conservation, and robustness of the proposed method.