Evaluation of the Least-Squares Method for the Solution of the Population Balance Equation

Spectral methods are evaluated for the solution of population balance problems. Both a simplified population balance equation (PBE) with an analytical solution available and a rigorous PBE are considered in this numerical analysis. In comparison with the orthogonal collocation, tau, and Galerkin methods, the least-squares method does not produce the same favorable results. On the other hand, the least-squares method with a direct minimization solver is capable of producing equally accurate results of the model equations as the orthogonal collocation, tau, and Galerkin methods. Compared to the conventional least-squares method, the direct minimization formulation is better conditioned but does not produce a symmetric positive-definite system matrix.