Information Technology Reference
In-Depth Information
Exercise 7.25.
Suppose we want to solve the diffusion equation in the limit where
@
u
=@t
!
0. This gives rise to the Poisson equation:
d
2
u
dx
2
D
f in Œ0; 1 :
There is no initial condition associated with this equation, because there is no evolu-
tion in time - the unknown function
u
is just a function of space:
u
D
u
.x/.However,
the equation is associated with boundary conditions, say,
u
.0/
D
u
.1/
D
0.
(a) Replace the second-derivative by a finite difference approximation. Explain
that this gives rise to a (tridiagonal) linear system, exactly as for the implicit
backward Euler scheme.
(b) Compare the equations from (a) with the equations generated by the backward
Euler scheme. Show that the former arises in the limit as t
!1
in the latter.
(c) Construct an analytical solution of the Poisson equation when f is constant.
(d) The result from (b) tells us that we can take one very long time step in a program
implementing the backward Euler scheme and then arrive at the solution of the
Poisson equation. Demonstrate, by using a program, that this is the case for the
test problem from (c).
˘
7.7
Projects
7.7.1
Diffusion of a Jump
We look at the physical problem of bringing two pieces of metal together, as
depicted in Fig.
7.4
on page
282
, where the initial temperatures of the pieces are
different. Heat conduction will smooth out the temperature difference, as shown in
Fig.
7.5
on page
282
.
(a) Formulate a one-dimensional initial-boundary value problem for this heat con-
duction application. The initial condition must reflect the temperature jump, and
the boundaries are insulated.
(b) Scale the problem. Hint: Use the same scales as we did for (
7.52
)-(
7.55
).
Many find modeling and scaling difficult, so the resulting scaled problem is
listed here for convenience:
@t
D
@
2
u
@
u
;
@x
2
@
u
@x
D
0; x
D
0; 1;
u
.x; 0/
D
0; x
0:5;
1; x > 0:5:
