  A new numerical scheme for conservation equations based on evolution by
asynchronous discrete events is presented. During each event of the scheme only
two cells of the underlying Cartesian grid are active, and an event is
processed as the exact evolution of this subsystem. This naturally leads to and
adaptive scheme in space and time. Numerical results are presented which show
that the error of the asynchronous scheme decreases to zero as a control
parameter is reduced. The construction of the scheme allows it to be expressed
as repeated multiplications of matrix exponentials on an initial state vector;
thus techniques such as the Goldberg series and the Baker Campbell Hausdorff
(BCH) formula can be used to explore the theoretical properties of the scheme.
We present the framework of a convergence proof in this manner.
