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the original form of the problem, with dimensions, but set the parameters in the
experiments such that one simulates a scaled form.
7.4
Explicit Numerical Methods
Our purpose now is to solve a one-dimensional diffusion equation, such as (7.1),
by means of a computer. To this end, we need to look at a mathematical problem
consisting of the diffusion PDE
and
boundary and initial conditions. Our goal is to
solve the (scaled) problem
k.x/
@
u
@x
C
f.x;t/;
%.x/c
v
.x/
@
u
@t
D
@
x
2
.0; 1/; t > 0;
(7.82)
@x
u
.0; t /
D
D
0
.t /;
t > 0;
(7.83)
@
@x
u
.1; t /
D
N
1
.t /;
t > 0;
(7.84)
u
.x; 0/
D
I.x/;
x
2
Œ0; 1 :
(7.85)
We refer to (
7.82
)-(
7.85
)asthe
continuous
problem. The PDE (
7.82
) is fulfilled
at all the
infinite
number of continuously distributed points, 0<x<1and t>0.
Moreover,
u
is defined at the same set of infinite points. Solving the problem (
7.82
)-
(
7.85
) on a computer requires us to construct an algorithm with a
finite
number of
steps for computing a
finite
number of parameters that describe
u
. This algorithm
must solve a
discrete
version of (
7.82
)-(
7.85
), and the process of constructing such
a discrete problem is called
discretization
.The
finite difference method
is one way
of discretizing problems involving PDEs.
7.4.1
The Basics of Finite Difference Discretizations
Applying the finite difference method to the problem (
7.82
)-(
7.85
) implies
(a) Constructing a
grid
, with a finite number of points in .x; t / space, see Fig.
7.11
(b) Requiring the PDE (
7.82
) to be satisfied at each point in the grid
(c) Replacing the derivatives by finite difference approximations
(d) Calculating (an approximation of)
u
at the grid points only
The finite difference discretization is characterized by these four steps. Requiring
the PDE to be satisfied at a number of discrete grid points, instead of a continuously
distributed set of .x; t /, represents an approximation. Replacing derivatives by finite
difference approximations is another source of error.
We will start with a version of the PDE (
7.82
)where%.x/, c
v
.x/,andk.x/ are
constant. These parameters are then “scaled away” such that the PDE reads
