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But, on the other hand, if we evaluate
something
at time t
D
t
nC1
,wegetthe
implicit definition of
v
nC1
:
v
nC1
D
v
n
C
t
something
.t
nC1
/;
which cannot be evaluated directly, because
something
may depend on
v
nC1
.Sothis
latter type is referred to as an implicit scheme. Generally speaking, implicit schemes
are often unconditionally stable but suffer from the fact that an equation has to be
solved, whereas explicit schemes are usually only conditionally stable but are very
simple to use.
Example 2.4.
Let us consider the initial value problem
y
0
.t /
D
y
2
.t /;
(2.34)
y.0/
D
1;
with the analytical solution
16
1
1
t
:
y.t/
D
An explicit scheme for this problem reads
y
nC1
y
n
t
D
y
n
;
so we get the explicit definition
y
nC1
D
y
n
C
ty
n
;
(2.35)
and thus we can easily compute the values of y
n
for n
D
1;2;:::.
Similarly, an implicit scheme reads
z
nC1
z
n
t
D
z
nC1
;
which can be rewritten as
z
nC1
t
z
nC1
D
z
n
:
Hence, if
z
n
is already computed, we can find
z
nC1
by solving the nonlinear equation
x
tx
2
D
z
n
(2.36)
and then setting
z
nC1
D
x: Since this is a simple second order polynomial equation,
we can solve it analytically and use one of the two solutions,
16
See Exercise
2.8
.
