It will be proven that an exponential tridiagonal difference scheme, when applied with a uniform mesh of size h to:
ε
u
x
x
+
b
(
x
)
u
x
=
f
(
x
)
\varepsilon {u_{xx}} + b(x){u_x} = f(x)
for
0
>
x
>
1
,
b
>
0
0 > x > 1,b > 0
, b and f smooth,
ε
\varepsilon
in (0, 1], and
u
(
0
)
u(0)
and
u
(
1
)
u(1)
given, is uniformly second-order accurate (i.e., the maximum of the errors at the grid points is bounded by
C
h
2
C{h^2}
with the constant C independent of h and
ε
\varepsilon
). This scheme was derived by El-Mistikawy and Werle by a
C
1
{C^1}
patching of a pair of piecewise constant coefficient approximate differential equations across a common grid point. The behavior of the approximate solution in between the grid points will be analyzed, and some numerical results will also be given.