We consider the approximate solution of the initial value problem
\[
∂
u
∂
t
=
∂
u
∂
x
,
u
(
x
,
0
)
=
v
(
x
)
,
\frac {{\partial u}}{{\partial t}} = \frac {{\partial u}}{{\partial x}},\quad u(x,0) = v(x),
\]
by a dissipative Galerkin method. When v is taken to have a jump discontinuity at zero, that discontinuity will propagate along
x
+
t
=
0
x + t = 0
, in the true solution u. Estimates in
L
2
{L_2}
and
L
∞
{L_\infty }
of the pollution effects of the discontinuity are found. These estimates show those effects to decay exponentially in
h
−
1
{h^{ - 1}}
in regions a fixed distance d from the discontinuity and exponentially in d for fixed h.