  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.
