The Diffusion Equation - Elements of Scientific Computing - page 356

leading) terms in the power expansion of e are proportional to t and x 2 .We
often write this as

.x 2 /
and say that the finite difference scheme is of first order in time (t 1 ) and second
order in space (x 2 ).
(c) From a specific solution (7.148) of a diffusion PDE, we realize that the values
of the solution decrease in time. Show that the exact solution (7.150)ofthe
corresponding numerical problem does not necessarily decrease in time; the
solution can increase if

e D .t/ C

O O

t > 1

2 x 2 :
Such an increase will be amplified with time, and we cannot allow this if the
numerical solution will have the same qualitative features as the solution of
the underlying PDE. We therefore say that the numerical solution is unstable
if t >

1
2

x 2 . A different way of achieving the same stability or instability
condition is given in the next chapter.

7.7.4 Compare Different Scalings

You are advised to work through Project 7.7.1 before attacking the present project.
We consider the same physical problem, but with different boundary conditions.
Let the boundary conditions be changed to a fixed temperature of U a at the left
boundary and known heat flow of Q b at the right boundary. The scaling of this
initial-boundary value problem was discussed in Sect. 7.3.5 . Two scales for the
unknown function were suggested, (7.73), referred to here as scale 1, and (7.79),
referred to as scale 2.

(a) Implement the scaled problem in the computer code from Project 7.7.1 .Intro-
duce an input parameter for switching between scale 1 and 2. Run problems
with ˇ D 1, ˇ 1 with scale 1, and ˇ 1 with scale 2. As commented
on in Sect. 7.3.5 , the latter two simulations demonstrate that the scaling fails, in
the sense that the magnitude of the solution is not about unity. The purpose of
this project is to analyze this observation a bit more by looking into analytical
expressions.
(b) As time increases, we expect the solution to stabilize and change little with time,
since there are no time-dependent boundary conditions or source terms in our
present model. Mathematically this means that we expect

@ u
@t D 0:

lim
t!1

In this limit, we have a stationary solution u s .x/ of the problem. The stationary
solution does not depend on the initial condition, i.e., the initial condition is