We categorize some of the finite-difference methods that can be used to treat the initial-value problem for the boundary-layer differential equation
\[
(
1
)
μ
y
′
=
f
(
y
,
x
)
;
y
(
0
)
=
y
0
.
(1){\text { }}\mu y’ = f(y,x);y(0) = {y^0}.
\]
These methods take the form
\[
(
2
)
∑
i
=
0
k
α
i
Y
n
+
i
=
h
1
−
γ
∑
i
=
0
k
β
i
f
(
Y
n
+
i
,
x
n
+
i
)
+
R
n
,
(2){\text { }}\sum \limits _{i = 0}^k {{\alpha _i}{Y_{n + i}} = {h^{1 - \gamma }}} \sum \limits _{i = 0}^k {{\beta _i}f} ({Y_{n + i}},{x_{n + i}}) + {R_n},
\]
where
α
ν
{\alpha _\nu }
and
β
ν
(
ν
=
0
,
1
,
⋯
,
k
)
{\beta _\nu }(\nu = 0,1, \cdots ,k)
denote real constants which do not depend upon
n
,
R
n
n,{R_n}
is the round-off error,
μ
=
h
r
,
0
>
γ
>
1
\mu = {h^r},0 > \gamma > 1
, and
h
h
is the mesh size. We define a new kind of stability called
μ
\mu
stability and prove that under certain conditions
μ
\mu
stability implies convergence of the difference method. We investigate
μ
\mu
stability and the optimal methods which it allows, i.e., methods of maximum accuracy. The idea of relating
μ
\mu
to
h
h
allows us to study the nature of the difference equation for very small
μ
\mu
. We can, however, look at this in another way. Given a differential equation in the form of Eq. (1) we ask how can we choose
h
h
so that the associated difference equation will give an accurate approximation. If
μ
\mu
is sufficiently small, choose
h
h
by the formula
h
=
μ
1
/
γ
h = {\mu ^{1/\gamma }}
where
0
>
γ
>
1
0 > \gamma > 1
.