(15d) Efficient Numerical Strategies for Multidimensional Population Balance Models and Transport Equations

Multidimensional population balance models (PBMs) can be used to describe many physical, chemical, and biological processes possessing a distribution over two or more intrinsic variables e.g., size and composition of a bubble. The descriptive capability of multidimensional PBMs have resulted in their application to systems containing cells, viruses, bubbles, and crystals, among others. Most multidimensional PBMs cannot be solved analytically, necessitating numerical methods for their solution which typically employ high-order finite difference or finite volume methods for accuracy. This presentation describes an upwind finite difference scheme employing operator splitting and variable transformations, which exploits the commutative property of the differential operators found in many classes of PBMs. When the scheme is employed at the limit of numerical stability, zero numerical discretization error occurs for certain classes of PBMs. Results are compared to high-order weighted essentially non-oscillatory (WENO) schemes in terms of accuracy and computational cost. The proposed scheme is very computationally efficient, and for some systems only requires memory reallocation. In addition to building on past research from the group, we explore and demonstrate the applicability of the strategies employed in the proposed scheme to the Advection-Diffusion-Reaction (ADR) transport equation, enabling the efficient and accurate solution of some classes of the ADR equation. Multiple case studies considering various classes of multidimensional PBMs and ADRs are presented to illustrate the proposed scheme’s performance in relation to other commonly employed schemes. Orders-of-magnitude reductions in computational cost compared to traditional finite difference and finite volume methods are demonstrated, while having higher numerical accuracy. Although the scheme does not apply to all partial differential equations, applicability is demonstrated for a large array of models that arise in chemical and biological systems. The low computational cost enables state estimation and model predictive control algorithms to be implemented in real time on processors with low computational power, while explicitly incorporating multidimensional partial differential equations into the computations.