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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. [ABSTRACT FROM AUTHOR]

In this paper, a detailed theoretical and experimental study on some computational aspects of high order discrete orthogonal Racah polynomials (RPs) and their corresponding moments is carried out. Initially, the numerical overflow problem related to RPs computation is solved by using modified recurrence relations of RPs with respect to the polynomial order n and the variable s. Moreover, the recursive nature of the resulting relations considerably accelerates the computation of RPs. Then, the problem of numerical errors propagation that occurs during the recursive computation of RPs is solved. Indeed, the proposed solution relies on the use of a new numerical method that detects unstable values and sets them to zero. This ensures the numerical stability of high-order RPs. Next, a fast method is presented to significantly reduce the required time for reconstructing large-size 1D signal. This method involves the transformation of a 1D signal into a 2D array, then using matrix reconstruction formulas in the 2D domain. The simulation and comparison results clearly show that the proposed computation methods are very useful for the fast and stable analysis of large-size signals and 2D/3D images by Racah moments (RMs). [ABSTRACT FROM AUTHOR]