Efficient numerical schemes for population balance models

Abstract

Population balance models (PBM) describe a wide array of physical, chemical, and biological processes having a distribution over some intrinsic property, and are used to model cells, viruses, aggregates, bubbles, and crystals. The ubiquity of PBMs motivates generalizable and accurate approaches for their numerical solution. Typically, high-order finite difference or finite volume methods are used. We propose a finite difference scheme at the limit of numerical stability that results in discretization error that is zero for certain classes of PBMs and low enough to be acceptable in other applications. The scheme employs specially constructed meshes and, in some cases, variable transformations. The scheme has very low computational cost – sometimes as low as memory reallocation with no floating point operations. Case studies are presented throughout that demonstrate the scheme’s performance in relation to other commonly employed schemes.