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(c) Write a computer program that solves the scaled problem by the explicit forward
Euler scheme.
(d) Show that
u
.x; t /
D
e
2
t
cos.x/
is a solution of the scaled problem if the initial condition is set to
u
.x; 0/
D
cos.x/. Implement this initial condition, run the code, and plot the maximum
error as a function of time. Argue why you think the code is free of bugs.
(e) Turn to the step function as the initial condition in the code. State clearly the
expected qualitative behavior of the evolution of the temperature, and comment
plots to provide confidence in the computed results.
(f) Find from computer experiments the time T it takes to reach a stationary
(constant) solution.
(g) You are now asked to plot the physical time T (with dimension) it takes to
achieve a constant temperature in the two metal pieces as a function of the heat
conduction coefficient and the total length of the pieces. You do not need to
produce the plot, but you need to explain how you will solve the problem. (Hint:
There is no need for simulations beyond the single run from (f) above! And
for those who are too accurate with the mathematics and give up because T is
infinite, remember that scientific computing is about approximations.)
7.7.2
Periodical Injection of Pollution
This project concerns the periodic injection of mass in a diffusive transport problem.
One important application can be a factory that launches pollution into the air in a
periodic manner. The goal of the study is to see how the pollution spreads in time
and how areas at some distance from the source are affected.
We assume that the pollution spreads equally in all three space dimensions. Since
the source of pollution, i.e. thepipe at the factory, is localized in a very small area,
it is reasonable to presume that the concentration depends on time and the distance
from the source. This means that the problem exhibits spherical symmetry, with the
origin at the source. In spherical coordinates the problem has only two independent
variables: time t and the radial distance r from the source.
The diffusion PDE expressed in spherical coordinates reads
r
2
@r
c.r; t/
C
f.r;t/:
@
@t
c.r; t/
D
k
1
r
2
@
(7.149)
(a) The source term f.r;t/ is used to model how pollution is injected into the air.
We assume that the factory is operative for 8 h/day. During operation, f
D
K,
where K is a constant reflecting the amount of pollution that enters the air per
unit time. The source f is also localized; during operation f
D
K in a small
