We describe a new numerical three-dimensional relativistic hydrodynamic code and present the results of three validation tests. A comparison of an axisymmetric jet simulation using the three-dimensional code, with corresponding results from an earlier two-dimensional code, reveal that ( 1) the enforcement of axisymmetry in the two-dimensional case had no significant influence on the global morphology and dynamics and ( 2) although three-dimensional studies typically have lower resolution than those using two dimensions, limiting their ability to fully capture internal jet structure, such three-dimensional studies can provide a reliable model of global morphology and dynamics. The three-dimensional code has been used to study the deflection and precession of relativistic flows. We find that even quite fast jets ( 10) can be significantly influenced by impinging on an oblique density gradient, exhibiting a rotation of the Mach disk in the jets head. The flow is bent via a potentially strong, oblique internal shock that arises owing to asymmetric perturbation of the flow by its cocoon. In extreme cases this cocoon can form a marginally relativistic flow orthogonal to the jet, leading to large-scale dynamics quite unlike that normally associated with astrophysical jets. Exploration of a gamma = 5 flow subject to a large amplitude precession (semiangle 11.degrees25) shows that it retains its integrity, with modest reduction in Lorentz factor and momentum flux, for almost 50 jet radii, but thereafter the collimated flow is disrupted. The flow is approximately ballistic, with velocity vectors not aligned with the local jet wall. However, sufficiently large changes in flow direction take place within the jet that for observers close to the jet axis, significant changes in Doppler boost would be evident along the flow. We consider simple estimators of the flow emissivity in each case and conclude that (1) while the oblique internal shocks that mediate a small change in the direction of the deflected flows have little impact on the global dynamics, significantly enhanced flow emission ( by a factor of 2-3) may be associated with such regions and ( 2) the convolution of rest-frame emissivity and Doppler boost in the case of the precessed jet invariably leads to a core-jet like structure but that intensity fluctuations in the jet cannot be uniquely associated with either change in internal conditions or Doppler boost alone but, in general, are a combination of both factors.
