Advanced approximation method with a convergence acceleration parameter for multidimensional aggregation model

This study presents a highly efficient and accurate approximate method based on a convergence acceleration parameter to approximate a nonlinear multidimensional aggregation population balance equation. Optimal tuning of the acceleration parameter significantly enhances solution quality over extended temporal domains and overcomes key limitations of existing approaches. Deeper mathematical insight is provided through a discussion of the existence of the proposed approach within the framework of a nonlinear aggregation model. Convergence analysis and error estimates are established using the fixed point theorem and the contractive mapping principle, thereby proving the existence of solutions to the aggregation model. The accuracy and efficiency of the proposed approach are demonstrated by computing approximate solutions for the number density function and its moments for physically relevant kernels. For analytically tractable kernels, results are validated against exact solutions. For complex size-dependent kernels, including polymerization, Ruckenstein–Pulvermacher, and shear kernels, the obtained results are compared with the existing finite volume scheme, homotopy analysis method, and optimal decomposition method. The results show that the proposed approach achieves higher accuracy in capturing number density functions and their integral moments while requiring significantly fewer series terms than existing methods.