# Efficient numerical schemes for population balance models # Efficient numerical schemes for population balance models

This article discusses the Efficient numerical schemes for population balance models. Using the wrong scheme to solve population balance equations can result in catastrophic errors, but using a suitable scheme can vastly improve your results. In this article, we’ll look at some efficient numerical schemes, including the explicit Euler scheme and the implicit Crank-Nicholson scheme, that you can use when solving population balance equations. First, however, let’s look at population balance equations and how they relate to population models in general.

#### A finite difference technique is presented that has zero numerical discretization mistakes.

The scheme guarantees that the first-order approximation, concerning time, of the solution of a two-dimensional population balance model is exact in that it satisfies an order of one ordinary differential equation. This finite difference scheme is based on the following three assumptions:

The time interval Δt is sufficiently small relative to all other parameters.

The rates at which individuals are produced and die are constant and proportional over any finite time interval Δt.

The generation times are independent of age.

#### The method is demonstrated for types of population equilibrium models.

1. It is well-known that the distribution of population density in a given region, as well as the corresponding level of spatial variation, can have a substantial impact on disease transmission. In this paper, we derive and implement efficient numerical schemes for population balance models with constant proportionate mixing, which are written in implicit form. The proposed methods are demonstrated by applying them to two illustrative examples:
An epidemiological model with seasonal variability
A stochastic epidemic model in an infinite homogeneous region. We also present empirical evidence on the efficiency of our approach.

#### The scheme employs specially constructed meshes and variable transformations.

The scheme we present is based on the idea that a variable in the model can be replaced by its average value. This scheme has been used before to solve variational problems but has not been applied to population balance models. We employ specially constructed meshes and variable transformations to obtain efficient numerical schemes that exhibit exponential convergence concerning time and space. Furthermore, the approach extends the method of moments for deriving conservation laws. Which are valid when there are no spatial correlations.

#### The scheme has a meagre computational cost, sometimes as low as memory reallocation.

This paper proposes an efficient numerical scheme for population balance models. The proposed scheme has a meagre computational cost, sometimes as low as memory reallocation. In addition, it does not require resizing the matrix or matrix inversion. The same is valid for solving linear systems arising from linearization. The result is also shown to be reliable by theoretical and practical analysis.

The efficiency and robustness of our method make it a promising alternative to more complex methods. Such as spectral projection and quasi-spectral methods when dealing with population balance equations. These are increasingly crucial in fluid dynamics simulations. Because they often arise in hydrodynamics, molecular dynamics, biophysics studies on liquids and gases, electron transport through semiconductors, etc.

In summary, the new approach presented here is highly efficient numerically while robust concerning noise and errors typical of real-life applications.

#### The system operates at the limit of numerical strength.

One of the main difficulties in solving population balance models is that they can become nonlinear. Due to the changing biomass of a species. One way to deal with this issue is by using efficient numerical schemes that are stable at the limit of numerical stability. Using these methods makes it possible to obtain solutions more accurately and in less time than other methods. These efficient numerical schemes have been used for many problems such as nitrogen fixation, phytoplankton growth and nitrification.

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