Consider the initial value problem (1.1)
\[
d
d
t
u
(
t
)
=
A
u
(
t
)
+
f
(
t
)
,
t
>
0
,
\frac {d}{{dt}}u(t) = Au(t) + f(t),\quad t > 0,
\]
(1.2)
\[
u
(
0
)
=
u
0
,
u(0) = {u_0},
\]
where A is a linear operator taking
D
(
A
)
⊂
X
D(A) \subset X
into X, where X is a Banach space. Consider also semidiscrete numerical methods of the form: find
U
N
(
t
)
:
[
0
,
T
]
→
X
N
{U_N}(t):[0,T] \to {X_N}
such that
(
1.1
′
)
(1.1\prime )
\[
d
U
N
d
t
=
A
N
U
N
+
P
N
f
,
\frac {{d{U_N}}}{{dt}} = {A_N}{U_N} + {P_N}f,
\]
(
1.2
′
)
(1.2\prime )
\[
U
N
(
0
)
=
U
N
0
∈
X
N
,
{U_N}(0) = U_N^0 \in {X_N},
\]
where
X
N
{X_N}
is a finite dimensional subspace and
P
N
{P_N}
is a projector onto
X
N
{X_N}
. The study of such numerical methods may be related to the approximation of semigroups and Laplace transform methods making use of the resolvent operators
(
A
−
λ
I
)
−
1
,
(
A
N
−
λ
I
N
)
−
1
{(A - \lambda I)^{ - 1}},{({A_N} - \lambda {I_N})^{ - 1}}
. The basic results require stability or weak stability and give convergence rates of the same order as in the steady state problems.