  This paper studies the physical-constraints-preserving (PCP) Lagrangian
finite volume schemes for one- and two-dimensional special relativistic
hydrodynamic (RHD) equations. First, the PCP property (i.e. preserving the
positivity of the rest-mass density and the pressure and the bound of the
velocity) is proved for the first-order accurate Lagrangian scheme with the
HLLC Riemann solver and forward Euler time discretization. The key is that the
intermediate states in the HLLC Riemann solver are shown to be admissible or
PCP when the HLLC wave speeds are estimated suitably. Then, the higher-order
accurate schemes are proposed by using the high-order accurate strong stability
preserving (SSP) time discretizations and the scaling PCP limiter as well as
the WENO reconstruction. Finally, several one- and two-dimensional numerical
experiments are conducted to demonstrate the accuracy and the effectiveness of
the PCP Lagrangian schemes in solving the special RHD problems involving strong
discontinuities, or large Lorentz factor, or low rest-mass density or low
pressure etc.
