A least-squares method with direct minimization for the solution of the breakage–coalescence population balance equation

A least-squares method with a direct minimization algorithm is introduced to solve the non-linear population balance equation that consists of both breakage and coalescence terms. The least-squares solver, direct minimization solver together with a finite difference solver are implemented for comparisons. It is shown that the coalescence term introduces a strong non-linear behavior which can affect the robustness of the numerical solvers. In the comparison with the least-squares method, the direct minimization method is proved to be capable of producing equally accurate results, while its formulation is better conditioned. In the case of a non-linear population balance equation system, the direct minimization method converges faster than the standard least-squares method.