We performed relativistic magnetohydrodynamic simulations of the hydrodynamic boosting mechanism for relativistic jets explored by Aloy & Rezzolla (2006) using the RAISHIN code. Simulation results show that the presence of a magnetic field changes the properties of the shock interface between the tenuous, overpressured jet ($V^z_j$) flowing tangentially to a dense external medium. Magnetic fields can lead to more efficient acceleration of the jet, in comparison to the pure-hydrodynamic case. A ``poloidal magnetic field ($B^z$), tangent to the interface and parallel to the jet flow, produces both a stronger outward moving shock and a stronger inward moving rarefaction wave. This leads to a large velocity component normal to the interface in addition to acceleration tangent to the interface, and the jet is thus accelerated to larger Lorentz factors than those obtained in the pure-hydrodynamic case. Likewise, a strong ``toroidal magnetic field ($B^y$), tangent to the interface but perpendicular to the jet flow, also leads to stronger acceleration tangent to the shock interface relative to the pure-hydrodynamic case. Overall, the acceleration efficiency in the ``poloidal case is less than that of the ``toroidal case but both geometries still result in higher Lorentz factors than the pure-hydrodynamic case. Thus, the presence and relative orientation of a magnetic field in relativistic jets can significant modify the hydrodynamic boost mechanism studied by Aloy & Rezzolla (2006).
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