An hp-adaptive quadrature method for irregular integrands: Application to the population balance equation birth term

Abstract

The solution of the population balance equation requires the integration of several source terms. In the numerical weighted residuals methods, Gaussian quadrature is a natural candidate for numerical integration. Previous works using the weighted residuals methods for solving the population balance equation did use a fixed grid of quadrature points. This work shows that the use of adaptive quadrature points for the numerical integration can lead to more efficient and accurate solutions of the equation. For cases where the integrand shows a high degree of irregularity, the hp-optimization method distributes the quadrature points such that the method becomes more efficient than with a fixed grid. An additional improvement is that the amount of quadrature points changes to fit the need for each integral present, rather than having one set of quadrature points for all cases. A simple population balance model demonstrates the use of the adaptive quadrature approach.