We consider the numerical solution of the stochastic partial differential equation
∂
u
/
∂
t
=
∂
2
u
/
∂
x
2
+
σ
(
u
)
W
˙
(
x
,
t
)
{\partial u}/{\partial t}={\partial ^2u}/{\partial x^2}+\sigma (u)\dot {W}(x,t)
, where
W
˙
\dot {W}
is space-time white noise, using finite differences. For this equation Gyöngy has obtained an estimate of the rate of convergence for a simple scheme, based on integrals of
W
˙
\dot {W}
over a rectangular grid. We investigate the extent to which this order of convergence can be improved, and find that better approximations are possible for the case of additive noise (
σ
(
u
)
=
1
\sigma (u)=1
) if we wish to estimate space averages of the solution rather than pointwise estimates, or if we are permitted to generate other functionals of the noise. But for multiplicative noise (
σ
(
u
)
=
u
\sigma (u)=u
) we show that no such improvements are possible.