Summary: Two new transport algorithms for solving the one-dimensional relativistic hydrodynamic equations of motion are discussed. One of them, relativistic HLLE, is based on the HLLE upwind scheme, while the other is a relativistic adaptation of some kinds of flux corrected transport algorithms. A comparison with other numerical methods used in relativistic hydrodynamics including fully implicit techniques is given. Emphasis is put on the dependence of the results on the relativistic gamma factor and on the adiabatic exponent entering the ideal gas equation of state. Future applications of the algorithms are particularly in the field of ultra-relativistic heavy ion collisions that require transport methods not critically dependent on the Lorentz gamma factor or the equation of state.