Numerical Modeling of Financially Sustainable Urban Wastewater Systems

Abstract

A system dynamics model first developed using the software Stella 7.0.2, which explores the complex behavior of the financially sustainable management of water distribution infrastructure, was converted here into a system of coupled non-linear algebraic differential equations (DAEs). Each differential equation involved a time derivative on a primary variable specifying the temporal evolution of the system. In addition, algebraic (secondary) equations and variables specified the non-linearity inherent in the system as well as any controls on the primary variables constraining the physical evolution of the system relevant to the problem at hand. While Stella employed a Runge-Kutta numerical strategy, the numerical DAE method used a fully-explicit, fully-implicit and Crank-Nicolson Euler scheme combined with a fixed-point iteration to resolve the non-linearity. The Runge-Kutta and numerical DAE solutions deviate markedly when the non-linearity of the system becomes pronounced. I demonstrate point-wise stability of the numerical DAE solution as the timestep is refined. Furthermore, the refined numerical DAE solution does not exhibit any of the spurious oscillations inherent in the Runge-Kutta solution and is physically correct for the problem at hand.