The full multicomponent population dynamic equation is developed for systems with coagulation and solved using the split composition distribution method. A full discussion of the origins of this method and its advantages over conventional methods is followed by a complete and rigorous explanation of its numerical implementation. The Wiener expansion is introduced as a method for representing compositional variation in a number density distribution that is solved by orthogonal collocation on finite elements. Results are obtained by numerical integration using fifth-order Runge-Kutta and favorably compared to analytical results.