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This paper studies the two-stage fourth-order accurate time discretization[J.Q.Li and Z.F.Du,SIAM J.Sci.Comput.,38 (2016)]and its application to the special relativistic hydrodynamical equations.Our analysis reveals that the new two-stage fourth-order accurate time discretizations can be proposed.With the aid of the direct Eulerian GRP (generalized Riemann problem) methods and the analytical resolution of the local “quasi 1D” GRP,the two-stage fourth-order accurate time discretizations are successfully implemented for the 1D and 2D special relativistic hydrodynamical equations.Several numerical experiments demonstrate the performance and accuracy as well as robustness of our schemes.