Diffusion Equations

Abstract

The famous diffusion equation, also known as the heat equation, reads @u @t D ˛ @ 2 u @x 2 ; where u.x; t/ is the unknown function to be solved for, x is a coordinate in space, and t is time. The coefficient ˛ is the diffusion coefficient and determines how fast u changes in time. A quick short form for the diffusion equation is u t D ˛u xx. Compared to the wave equation, u t t D c 2 u xx , which looks very similar, the diffusion equation features solutions that are very different from those of the wave equation. Also, the diffusion equation makes quite different demands to the numerical methods. Typical diffusion problems may experience rapid change in the very beginning, but then the evolution of u becomes slower and slower. The solution is usually very smooth, and after some time, one cannot recognize the initial shape of u. This is in sharp contrast to solutions of the wave equation where the initial shape is preserved in homogeneous media-the solution is then basically a moving initial condition. The standard wave equation u t t D c 2 u xx has solutions that propagate with speed c forever, without changing shape, while the diffusion equation converges to a stationary solution N u.x/ as t ! 1. In this limit, u t D 0, and N u is governed by N u 00 .x/ D 0. This stationary limit of the diffusion equation is called the Laplace equation and arises in a very wide range of applications throughout the sciences. It is possible to solve for u.x; t/ using an explicit scheme, as we do in Sect. 3.1, but the time step restrictions soon become much less favorable than for an explicit scheme applied to the wave equation. And of more importance, since the solution u of the diffusion equation is very smooth and changes slowly, small time steps are not convenient and not required by accuracy as the diffusion process converges to a stationary state. Therefore, implicit schemes (as described in Sect. 3.2) are popular, but these require solutions of systems of algebraic equations. We shall use ready-made software for this purpose, but also program some simple iterative methods. The exposition is, as usual in this book, very basic and focuses on the basic ideas and how to implement. More comprehensive mathematical treatments and classical analysis of the methods are found in lots of textbooks. A favorite of ours in this respect is the one by LeVeque [13]. The books by Strikwerda [17] and by Lapidus and Pinder [12] are also highly recommended as additional material on the topic.