  We present a new open-source axisymmetric general relativistic hydrodynamics
code Gmunu (General-relativistic multigrid numerical solver) which uses a
multigrid method to solve the elliptic metric equations in the conformally flat
condition (CFC) approximation on a spherical grid. Most of the existing
relativistic hydrodynamics codes are based on formulations which rely on a
free-evolution approach of numerical relativity, where the metric variables are
determined by hyperbolic equations without enforcing the constraint equations
in the evolution. On the other hand, although a fully constrained-evolution
formulation is theoretical more appealing and should lead to more stable and
accurate simulations, such an approach is not widely used because solving the
elliptic-type constraint equations during the evolution is in general more
computationally expensive than hyperbolic free-evolution schemes. Multigrid
methods solve differential equations with a hierarchy of discretizations and
its computational cost is generally lower than other methods such as direct
methods, relaxation methods, successive over-relaxation. With multigrid
acceleration, one can solve the metric equations on a comparable time scale as
solving the hydrodynamics equations. This would potentially make a fully
constrained-evolution formulation more affordable in numerical relativity
simulations. As a first step to assess the performance and robustness of
multigrid methods in relativistic simulations, we develop a hydrodynamics code
that makes use of standard finite-volume methods coupled with a multigrid
metric solver to solve the Einstein equations in the CFC approximation. In this
paper, we present the methodology and implementation of our code Gmunu and its
properties and performance in some benchmarking relativistic hydrodynamics
problems.
