High-Order Compact Difference Method for Solving Population Balance Equations in Batch Crystallization

As there is typically no analytical solution to most population balance equations (PBEs) of interest, computationally expensive high-order or high-resolution methods are typically used to obtain accurate numerical solutions. In this study, a new high-order compact difference (HOCD) method is proposed to solve the PBE with fourth-order accuracy in both space and time. This method provides high computational accuracy with a computationally efficient solution for one-dimensional population balance modeling in batch cooling crystallization processes. A compact difference scheme is proposed based on a two-layer format, with three grid points involved at each time level. Tridiagonal linear equations are solved directly using Tomas’ algorithm. Stability is demonstrated through von Neumann stability analysis. Compared to the Upwind, Lax–Wendroff, and high-resolution finite volume (HR-FVM) methods, the HOCD method offers higher computational accuracy and efficiency, without numerical diffusion or dispersion. The effectiveness of this method is demonstrated through multiple case studies.