Information Technology Reference
In-Depth Information
Chapter 7
The Diffusion Equation
This chapter treats the numerical simulation of diffusion processes. Diffusion takes
place in space and time simultaneously and is an important phenomenon in nature
and technology. Scientific computations of physical quantities such as temperature,
pollution, and velocity are frequently based on models for diffusion. Diffusion is
often coupled with other processes in physical problems, but in this chapter we will
only consider diffusion as an isolated process.
A central message to communicate in the present chapter is that diffusion arises
in widely different physical contexts, and the derivations of appropriate mathemat-
ical models differ correspondingly, but all the models we look at can be expressed
in the same form. Therefore we can derive common algorithms for simulating dif-
ferent diffusion processes, and implement these algorithms in a program that can be
executed to study various diffusion phenomena in different physical contexts. For
example, we can use the program to study both heat conduction and the diffusion of
bacteria. What we learn about heat conduction can be transferred to knowledge on
how bacteria populations move in space and time.
The idea that the same mathematical model can describe different kinds of
physics is quite simple, but it is non-trivial to get all the details right such that
you can really write a single program for a collection of application areas, run the
program, and present results in the context of a particular physical case. That is why
we explain all the details regarding how a single model and program can be used for
three widely differing physical applications: heat conduction, diffusive (molecular)
transport, and thin-film fluid flow.
The main purpose of the chapter is to learn about mathematical models involv-
ing partial differential equations (PDEs). The PDE arising in diffusion phenomena
is one of the simplest PDEs, but to formulate and solve it, we encounter the major
topics about PDEs and the numerical methods used to solve them. To keep the math-
ematical and numerical details at a minimum, we mainly work with one-dimensional
diffusion models.
Sections 7.1.1 - 7.1.4 explain the basic physical features of diffusion processes
with examples from heat conduction and diffusive transport (fluid mixing). We also
argue why one-dimensional models are relevant in a three-dimensional world. Sec-
tion 7.2 is of particular importance since here we state the governing PDE model for
diffusion and introduce basic mathematical quantities.
Search WWH ::




Custom Search