Application of orthogonal collocation in aerosol science: Fast calculation of the coagulation tensor

The ability of spectral methods to solve population balances depends on the smoothness of the PDF, determined by the position and width of the region of interest and by the shape of the PDF. The use of adaptive schemes based on appropriate time-dependent mappings and domain decompositions can overcome that limitation. However, these adaptive schemes may force the calculation of the coagulation tensor at every time step, involving N3+N2N3+N2 integrals for N spectral components.A new fast algorithm that enables the calculation of the coagulation tensor based on N integrals (which do not depend on the coagulation kernel) is introduced. This new algorithm allows for a great reduction in the time needed to calculate the coagulation tensor, enabling the efficient implementation of adaptive spectral methods for aerosol problems involving coagulation.