High resolution compact implicit numerical scheme for conservation laws

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