  This paper extends the second-order accurate BGK finite volume schemes for
the ultra-relativistic flow simulations [5] to the 1D and 2D special
relativistic hydrodynamics with the Synge equation of state. It is shown that
such 2D schemes are very time-consuming due to the moment integrals (triple
integrals) so that they are no longer practical. In view of this, the
simplified BGK (sBGK) schemes are presented by removing some terms in the
approximate nonequilibrium distribution at the cell interface for the BGK
scheme without loss of accuracy. They are practical because the moment
integrals of the approximate distribution can be reduced to the single
integrals by some coordinate transformations. The relations between the left
and right states of the shock wave, rarefaction wave, and contact discontinuity
are also discussed, so that the exact solution of the 1D Riemann problem could
be derived and used for the numerical comparisons. Several numerical
experiments are conducted to demonstrate that the proposed gas-kinetic schemes
are accurate and stable. A comparison of the sBGK schemes with the BGK scheme
in one dimension shows that the former performs almost the same as the latter
in terms of the accuracy and resolution, but is much more efficiency.
