
PART II — FOUNDATIONS OF MODELING


Dynamical processes are the result of the storage, flow, and production of quantities such as water, money, angular momentum, antelopes, natural gas, heat, and many more. This image can be easily translated into powerful mathematical statements—socalled laws of balance. A law of balance expresses how the content of a storage unit changes as the result of flow and production processes. If the laws of balance are combined with special laws for the currents and production rates, models of dynamical processes are created.
In Chapter 3, we will investigate laws of balance in some detail. How do they arise? What are their elements? How can we describe processes and storage? We will learn to formulate, to write, and to transform the most general form of a law of balance. Even though this is a mathematical business, it can be presented in very simple language, requiring little beyond arithmetic, algebra as well as interpretation and manipulation of graphs. A little familiarity with calculus won’t hurt, but you can actually take the subject of Chapter 3 as an informal and motivating introduction to the mathematics of dynamical systems. After laws of balance we will briefly study constitutive laws and, finally, the overall mathematical structure of models of uniform dynamical systems.
A modest amount of calculus will be required when we discuss what mathematics can contribute toward a qualitative understanding of dynamical system behavior in Chapter 4. Still, it may surprise you how much we can learn about dynamical systems without solving complicated equations in detail. Chapters 3 and 4 demonstrate the power of mathematics as a language for formulating and for discussing system dynamics models.
The solving of the equations that comprise a model commonly requires numerical methods. Analytical solutions are hard to come by, and usually we can find them only for the simplest of models. Therefore we will study some introductory aspects of numerical methods in the final fifth chapter. Again, we will be able to understand the major issues—stability and accuracy of numerical schemes—with a minimum of formal mathematics by investigating some of the simple algorithms.
The subject of this book are uniform dynamical systems which lead to socalled initial value problems (IVPs). Just to give you a feeling for how modeling and numerical methods are extended to the spatially continuous case, we will include some aspects of boundary value problems in the discussion in Chapters 3 and 5. Boundary value problems arise when we look not only at the time evolution of systems, but also their spatial structure. You will see how the ideas formed in the previous chapters and sections lend themselves to a treatment of this important field of application, which is at the heart of the socalled finite element models.
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