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Numerical simulation of solutions and moments of the smoluchowski coagulation equation

Researchers have employed variations of the Smoluchowski coagulation equation to model a wide variety of both organic and inorganic phenomena and with relatively few known analytical solutions, numerical solutions play an important role in studying this equation. In this article, we consider numerical approximations, focusing on how different discretization schemes impact the accuracy of approximate solution moments. Pursuing the eventual goal of comparing simulated solutions to experimental data, we must carefully choose the numerical method most appropriate to the type of data we attain. Within this context, we compare and contrast the accuracy and computational cost of a finite element approach and a finite volume-based scheme. Our study provides theoretical and numerical evidence that the finite element approach achieves much more accuracy when the system aggregates slowly, and it does so with much less computation cost. Conversely, the finite volume method is slightly more accurate approximating the zeroth moment when the system aggregates quickly and is much more accurate approximating the first moment in general. Lastly, our study also provides numerical evidence that the finite element method (conventionally considered first order) actually belongs to a class of discontinuous Galerkin methods that exhibit superconvergence, or second order in our case.