Introduction to Environmental Modeling

Introduction

We have worked through a number of equations involving dynamics of population, mass conservation, CSTRs, stream flow, and estuary flow. In many cases, problems described by differential equations are not directly solvable and must be approximated as discrete equations. In Section 7.3, we did some analysis on a discrete equation to come up with a mathematical condition for linear stability. In Chapter 16, we encountered unstable behavior in solving some discrete approximations of the convection dispersion equation. When these instabilities arise, the discrete solution to the equation “blows up” such that the numerical results produced are clearly nonsense (e.g. the dreaded result “NaN” is calculated). Although we have a mathematical theorem that indicates when a discrete equation will become unstable, it may be interesting to look at a physical problem to try to predict instability on that basis. For the cases considered here, various discrete forms of the differential equation describing contamination in a wellmixed lake will be examined. Although all the forms used are approximations to the differential equation, and reduce to the differential equation in the limit of a small time step, we will see that the discrete forms correspond to particular ways the system is operated. By identifying the modes of operation that a discrete equation describes, we can discover the stability properties of the equation. Some of these modes of operation are physically unacceptable, and this translates to instability in the discrete equation. When modes of operation are physically acceptable, the discrete equation is stable.

With the same objective of relating conservation equations to descriptions of physical processes that occur, here we relate mathematical behavior of discrete solutions to physical processes. The examples here are relatively simple in that they apply to processes described by an ordinary differential equation. Making the same analogy between physical processes and their mathematical description in other contexts is more difficult. Nevertheless, the fact that seemingly reasonable discrete approximations of differential equations can behave badly is an important issue to ponder.