The hydrodynamics of a system of conservation laws can often be understood by studying solutions to a Riemann problem, i.e., the evolution of step initial data in a hyperbolic system. When dispersion compensates for the large gradients induced by this initial data, wave-breaking is often resolved by a dispersive shock wave (DSW). An active area of research is the investigation of quantitative features of DSWs in continuum models arising in physical applied mathematics. This talk leverages modern techniques to investigate solutions of the Riemann problem for a semi-discrete conservation law. The semi-discrete model is obtained by applying a first-order centered difference scheme to the spatial derivative of the Hopf equation. Solutions to the Riemann problem reveal a surprisingly elaborate set of solutions to this example system. In addition to discrete analogs of well-known dispersive hydrodynamic solutions—rarefaction waves (RWs) and DSWs—additional unsteady solution families and finite-time blow-up are observed and characterized. We will also compare the dynamics of the Riemann problem to an integrable discretization of the spatial derivative.