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Efficient Numerical Schemes for Multidimensional Population Balance Models

Pavan Inguva, Richard D. Braatz

TL;DR

This work tackles the computational burden of solving multidimensional population balance models by leveraging operator splitting and problem-specific variable transformations that exploit commuting differential operators. By recasting PBMs into sequences of accurate 1D subproblems and operating at the stability limit ($CFL=1$), the method achieves high accuracy with minimal memory overhead, often matching or exceeding high-order solvers on challenging cases. The authors present a hierarchy of cases with constant, time-dependent, and nonhomogeneous growth rates, detailing exact transformations and several solver strategies, including upwind, WENO, and exact-splitting variants. The results demonstrate substantial diffusion reduction and, in many cases, machine-precision accuracy, with practical implications for online optimization and control of processes described by PBMs. Code availability further enables widespread adoption and benchmarking across dimensions and problem classes.

Abstract

Multidimensional population balance models (PBMs) describe chemical and biological processes having a distribution over two or more intrinsic properties (such as size and age, or two independent spatial variables). The incorporation of additional intrinsic variables into a PBM improves its descriptive capability and can be necessary to capture specific features of interest. As most PBMs of interest cannot be solved analytically, computationally expensive high-order finite difference or finite volume methods are frequently used to obtain an accurate numerical solution. We propose a finite difference scheme based on operator splitting and solving each sub-problem at the limit of numerical stability that achieves a discretization error that is zero for certain classes of PBMs and low enough to be acceptable for other classes. In conjunction to employing specially constructed meshes and variable transformations, the scheme exploits the commutative property of the differential operators present in many classes of PBMs. The scheme has very low computational cost -- potentially as low as just memory reallocation. Multiple case studies demonstrate the performance of the proposed scheme.

Efficient Numerical Schemes for Multidimensional Population Balance Models

TL;DR

This work tackles the computational burden of solving multidimensional population balance models by leveraging operator splitting and problem-specific variable transformations that exploit commuting differential operators. By recasting PBMs into sequences of accurate 1D subproblems and operating at the stability limit (), the method achieves high accuracy with minimal memory overhead, often matching or exceeding high-order solvers on challenging cases. The authors present a hierarchy of cases with constant, time-dependent, and nonhomogeneous growth rates, detailing exact transformations and several solver strategies, including upwind, WENO, and exact-splitting variants. The results demonstrate substantial diffusion reduction and, in many cases, machine-precision accuracy, with practical implications for online optimization and control of processes described by PBMs. Code availability further enables widespread adoption and benchmarking across dimensions and problem classes.

Abstract

Multidimensional population balance models (PBMs) describe chemical and biological processes having a distribution over two or more intrinsic properties (such as size and age, or two independent spatial variables). The incorporation of additional intrinsic variables into a PBM improves its descriptive capability and can be necessary to capture specific features of interest. As most PBMs of interest cannot be solved analytically, computationally expensive high-order finite difference or finite volume methods are frequently used to obtain an accurate numerical solution. We propose a finite difference scheme based on operator splitting and solving each sub-problem at the limit of numerical stability that achieves a discretization error that is zero for certain classes of PBMs and low enough to be acceptable for other classes. In conjunction to employing specially constructed meshes and variable transformations, the scheme exploits the commutative property of the differential operators present in many classes of PBMs. The scheme has very low computational cost -- potentially as low as just memory reallocation. Multiple case studies demonstrate the performance of the proposed scheme.
Paper Structure (20 sections, 68 equations, 11 figures, 1 table)

This paper contains 20 sections, 68 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Theory and Methods
  3. Case 1: PBMs with Constant Growth Rate
  4. Case 2: PBMs with Growth Rate Dependent on Intrinsic Variables Only
  5. PBMs with Growth Rate $G_{i} = G_{i}(a_{i})$
  6. PBMs with Growth Rate $G_{i} = G_{i}(\mathbf{a})$
  7. Case 3: PBMs with Time-Dependent Growth Rates
  8. PBMs with Growth Rate $G_{i} = G_{i}(t)$
  9. PBMs with Growth Rate $G_{i} = G_{i,t}(t)G_{i,a_{i}}(a_{i})$
  10. Case 4: Nonhomogeneous PBMs
  11. Nonhomogeneous PBMs with Constant Growth Rates
  12. PBMs with a Linear Nonhomogeneous Term
  13. Results and Discussion
  14. Case 1: PBMs with Constant Growth Rates
  15. Case 2: PBMs with Growth Rate $G_{i} = G_{i}(a_{i})$
  16. ...and 5 more sections

Figures (11)

  • Figure 1: Exemplar jagged mesh generated as part of the "Split-Trans-Nonuniform,Upwind" scheme.
  • Figure 2: Simulation results for Case 1 at $t=1.0$ using the various schemes. 101 grid points are used in both the $a_{1}$ and $a_{2}$ directions for the Upwind and Exact schemes while 100 cells in both the $a_{1}$ and $a_{2}$ directions for the WENO scheme.
  • Figure 3: Error analysis for Case 1. The use of dimensional splitting and employing $\text{CFL} = 1$ for each subproblem solves the PBM to machine precision as expected.
  • Figure 4: Simulation results for Case 2 at $t=1.0$ using the various schemes. 101 grid points are used in both the $a_{1}$ and $a_{2}$ directions for the Upwind schemes on a uniform grid and Exact schemes while 100 cells in both the $a_{1}$ and $a_{2}$ directions for the WENO scheme. The simulations on a nonuniform grid have 277 and 56 grid points in the $a_{1}$ and $a_{2}$ directions respectively.
  • Figure 5: Error analysis for Case 2.
  • ...and 6 more figures