We describe a method for the numerical solution of the relativistic Euler equations which we have found to be both robust and efficient, and which has enabled us to simulate relativistic jets. The technique employs a solver of the Godunov-type, with approximate solution of the local Riemann problems, applied to laboratory frame variables. Lorentz transformations provide the rest frame quantities needed for the estimation of wave speeds, etc. This is applied within the framework of an adaptive mesh refinement algorithm, allowing us to perform high-resolution, 2-D simulations with modest computing resources. We present the results of nonrelativistic, and relativistic ($��=5$ and $10$) runs, for adiabatic indices of $5/3$ and $4/3$. We find the same gross morphology in all cases, but the relativistic runs exhibit little instability and less well-defined structure internal to the jet: this might explain the difference between (relatively slow) BL~Lacs and (faster) QSOs. We find that the choice of adiabatic index makes a small but discernible difference to the structure of the shocked jet and ambient media.

11 pages, AAS LaTeX v3.0, (figure available from hughes@astro.lsa.umich.edu)