  We present a novel implicit scheme for the numerical solution of
time-dependent conservation laws. The core idea of the presented method is to
exploit and approximate the mixed spatial-temporal derivative of the solution
that occurs naturally when deriving some second order accurate schemes in time.
Such an approach is introduced in the context of the Lax-Wendroff (or
Cauchy-Kowalevski) procedure when the second time derivative is not completely
replaced by space derivatives using the PDE, but the mixed derivative is kept.
If approximated in a suitable way, the resulting compact implicit scheme
produces algebraic systems that have a more convenient structure than the
systems derived by fully implicit schemes. We derive a high resolution TVD form
of the implicit scheme for some representative hyperbolic equations in the
one-dimensional case, including illustrative numerical experiments.
