  The Riemann problem for the discrete conservation law $2 \dot{u}_n +
u^2_{n+1} - u^2_{n-1} = 0$ is classified using Whitham modulation theory, a
quasi-continuum approximation, and numerical simulations. A surprisingly
elaborate set of solutions to this simple discrete regularization of the
inviscid Burgers' equation is obtained. In addition to discrete analogues of
well-known dispersive hydrodynamic solutions -- rarefaction waves (RWs) and
dispersive shock waves (DSWs) -- additional unsteady solution families and
finite time blow-up are observed. Two solution types exhibit no known
conservative continuum correlates: (i) a counterpropagating DSW and RW solution
separated by a symmetric, stationary shock and (ii) an unsteady shock emitting
two counter-propagating periodic wavetrains with the same frequency connected
to a partial DSW or a RW. Another class of solutions called traveling DSWs,
(iii), consists of a partial DSW connected to a traveling wave comprised of a
periodic wavetrain with a rapid transition to a constant. Portions of solutions
(ii) and (iii) are interpreted as shock solutions of the Whitham modulation
equations.
