Numerical DAE Approach for Solving a System Dynamics Problem

Abstract

A system dynamics model first developed using modeling and simulation software that explores the complex behavior of the financially sustainable management of water distribution infrastructure was converted into a system of coupled nonlinear 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 nonlinearity 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. The objective of this exercise was to demonstrate that spurious oscillations in the modeling and simulation software solution are numerical aberrations. Furthermore, the numerical DAE solution is absent these same oscillations, exhibits point-wise stability, and converges to the physically correct solution. While the modeling and simulation software employed a fourth-order Runge-Kutta and first-order Euler 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 nonlinearity. The Runge-Kutta and numerical DAE solutions deviate markedly when the nonlinearity of the system becomes pronounced. Specifically, spurious oscillations in the numerical DAE solution disappear as the time step is refined. In contrast, they remain for the Runge-Kutta solution. The DAE solution is point-wise stable as the time step is refined and hence is physically correct. The broader impact of clarifying this type of behavior is to motivate the consideration of a DAE solution, when merited, by system dynamics modelers in civil engineering who are not experts in numerical methods.