  The paper proposes a second-order accurate direct Eulerian generalized
Riemann problem (GRP) scheme for the spherically symmetric general relativistic
hydrodynamical (RHD) equations and a second-order accurate discretization for
the spherically symmetric Einstein (SSE) equations. The former is directly
using the Riemann invariants and the Runkine-Hugoniot jump conditions to
analytically resolve the left and right nonlinear waves of the local GRP in the
Eulerian formulation together with the local change of the metrics to obtain
the limiting values of the time derivatives of the conservative variables along
the cell interface and the numerical flux for the GRP scheme. While the latter
utilizes the energy-momentum tensor obtained in the GRP solver to evaluate the
fluid variables in the SSE equations and keeps the continuity of the metrics at
the cell interfaces. Several numerical experiments show that the GRP scheme can
achieve second-order accuracy and high resolution, and is effective for
spherically symmetric general RHD problems.
