The Diffusion Equation - Elements of Scientific Computing - page 354

area r r 0 ,wherer 0 is typically the radius of the factory pipe. When the
factory is not in operation, no pollution enters the air (f D 0).
Find a mathematical specification of the function f.r;t/. Implement f as a
function in a program. (Hint: Exercise 7.1 provides some ideas.)
(b) At distances far from the source (r D 0) we assume that there is no pollution.
Mathematically, we can express this as

lim
r!1

c.r; t/ D 0:

Infinite domains are inconvenient in computer models, so we implement the
condition at a distance r D L,whereL is sufficiently large. Initially, we assume
that there is no pollution in the air. Set up a complete initial-boundary value
problem for this physical application.
(c) Construct an explicit finite difference scheme for (7.149). Hint: Use the ideas
from Sect. 7.4.7 .
(d) Set up a complete algorithm for computing c.r; t/.
(e) Implement the algorithm from (d) above in a computer program.
(f) Suggest a series of test problems with the purpose of checking that the program
implementation is correct.
(g) Choosing values for the constants K, k, L, r 0 , t , and the total simulation
time can be challenging. Scaling will help to make the choices of input data
significantly simpler. Scale the problem, using t c D r 0 =k as the time scale, r 0

as the space scale, and K as the scale for f . Show that the scaled PDE takes the
form (dropping the bars in scaled quantities)

r 2 C ˛f .r; t /;

@c

@t D 1

@
@r

@c
@r r 2

where ˛ is a dimensionless constant that equals unity if we choose the scale
for the concentration as t c K. Set up the complete scaled problem, and explain
how you can use the program developed for the unscaled problem by setting the
input parameters K, k, L, and so on, to suitable values.
(h) Perform numerical experiments to determine an appropriate location of the
“inifinite” boundary (L=r 0 ). The principle is that the solution should be approx-
imately unaffected by moving the boundary further toward infinity.
(i) Create an animation of the scaled concentration. The animation should contain
at least ten periods (days) of simulation.
(j) Improve the animation by plotting both the f and c functions such that one can
see the effect of the pollution injection on c.

(k) Discuss the validity of this statement: “The movie created in (j) is the complete
solution of the problem”.