Fast algorithms for hp-discretized univariate population balance aggregation integrals

The efficient numerical simulation of population balance equations requires sophisticated techniques in order to combine accuracy with efficiency. We will focus on the numerical treatment of aggregation integrals that often dominate the overall time in population balance simulations. Following a finite element approach, the density distribution is discretized through a piecewise polynomial of order p > 0 on a nested grid that is refined locally toward an arbitrary point. The proposed method conserves mass while reducing the quadratic complexity (in the dimension of the solution space) of the direct computation to an almost linear complexity. The complexity improvement is based on recursion formulas exploiting orthogonality properties of basis functions along with FFT on locally equidistant portions of the grid. We present numerical results for various initial conditions and provide heuristic criteria for the choice of polynomial degree and grid refinement.