We consider difference approximations to the model hyperbolic equation
u
t
=
u
x
{u_{t}} = {u_x}
which compute each new value
U
(
x
,
t
+
Δ
t
)
U(x,t + \Delta t)
as a combination of the known values
U
(
x
−
r
Δ
x
,
t
)
,
…
,
U
(
x
+
s
Δ
x
,
Δ
t
)
U(x - r\Delta x,t),\ldots ,U(x + s\Delta x,\Delta t)
. For such schemes we find the optimal order of accuracy: stability is possible for small
Δ
t
/
Δ
x
\Delta t/\Delta x
if and only if
p
⩽
min
{
r
+
s
,
2
r
+
2
,
2
s
}
p \leqslant \min \{ {r + s,2r + 2,2s} \}
. A similar bound is established for implicit methods. In this case the most accurate schemes are based on Padé approximations
P
(
z
)
/
Q
(
z
)
P(z)/Q(z)
to
z
λ
{z^\lambda }
near
z
=
1
z = 1
, and we find an expression for the difference
|
Q
|
2
−
|
P
|
2
|Q{|^2} - |P{|^2}
; this allows us to test the von Neumann condition
|
P
/
Q
|
⩽
1
|P/Q| \leqslant 1
. We also determine the number of zeros of
Q
Q
in the unit circle, which decides whether the implicit part is uniformly invertible.