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.

中文翻译：

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Gmunu：面向广义相对论流体动力学模拟的基于多重网格的爱因斯坦场方程求解器

我们提出了一种新的开源轴对称广义相对论流体动力学代码 Gmunu（广义相对论多重网格数值求解器），它使用多重网格方法来求解球形网格上保形平面条件 (CFC) 近似下的椭圆度量方程。大多数现有的相对论流体动力学代码都基于依赖于数值相对论的自由演化方法的公式，其中度量变量由双曲方程确定，而不强制执行演化中的约束方程。另一方面，尽管完全约束演化公式在理论上更具吸引力，并且应该会导致更稳定和准确的模拟，这种方法没有被广泛使用，因为在演化过程中求解椭圆型约束方程通常比双曲自由演化方案在计算上更昂贵。多重网格法求解具有离散化层次的微分方程，其计算成本通常低于直接法、松弛法、连续超松弛法等其他方法。使用多重网格加速，人们可以在与求解流体动力学方程相当的时间尺度上求解度量方程。这可能会使完全约束演化公式在数值相对论模拟中更加经济实惠。作为评估多重网格方法在相对论模拟中的性能和稳健性的第一步，我们开发了一个流体动力学代码，该代码利用标准有限体积方法和多重网格度量求解器来求解 CFC 近似中的爱因斯坦方程。在本文中，我们介绍了我们的代码 Gmunu 的方法和实现及其在一些基准相对论流体动力学问题中的属性和性能。