Information Technology Reference
In-Depth Information
Let us consider the initial value problem
y
0
.t /
D
100y.t/;
y.0/
D
1;
which gave us some difficulties above. Since a
D
100; we have
D
n
1
1
C
100t
y
n
D
N
N
C
100
n
;
which is easily seen to yield uniformly positive solutions; the oscillatory behavior
observed above is not present here. Note, in particular, that at time t
N
D
T
D
1; we
have
D
N
N
C
100
N
:
y
N
From the table below, we observe that this formula provides a fair approximation of
the exact solution.
Ny
N
10
1
3:85
10
11
10
2
7:89
10
31
10
3
4:05
10
42
10
7
3:72
10
44
Note that the exact solution is e
100
3:72
10
44
:
2.2.5
Explicit and Implicit Schemes
The scheme obtained here is
implicit
and it is also
unconditionally stable
.Sowhat
is the difference between an explicit and an implicit scheme? Suppose we have an
equation of the form
v
0
.t /
D
something
, (2.33)
and suppose also that we derive the numerical method by replacing the term
v
0
.t /
with a term of the form
v
nC1
v
n
t
:
Now, if we replace the right hand side of (2.33)by
something
evaluated at time
t
D
t
n
; then the scheme is called explicit and the reason is simply that we get an
explicit formula for
v
nC1
I
in fact,
v
nC1
D
v
n
C
t
something
.t
n
/:
