  Typically when a semi-discrete approximation to a partial differential
equation (PDE) is constructed a discretization of the spatial operator with a
truncation error $\tau$ is derived. This discrete operator should be
semi-bounded for the scheme to be stable. Under these conditions, the
Lax--Ricchtmyer equivalence theorem assures that the scheme converges and that
the error will be, at most, of the order of $\| \tau \|$. In most cases, the
error is in indeed of the order of $\| \tau \|$.
  We demonstrate that for the Heat equation stable schemes can be constructed,
whose truncation errors are $\tau$, however, the actual errors are much
smaller. This gives more degrees of freedom in the design of schemes which can
make them more efficient (more accurate or compact) than standard schemes. In
some cases, the accuracy of the schemes can be further enhanced using
post-processing procedures.
