Gmunu：基于多重网格的广义相对论流体力学爱因斯坦场方程求解器
Gmunu: Toward multigrid based Einstein field equations solver for general-relativistic hydrodynamics simulations
摘要：
我们提出了一个新的开源的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解决椭圆形度量方程的共形平坦条件（CFC）近似球网格。现有的相对论流体动力学代码的基础上的配方依赖于一个自由的数值相对论的方法，其中的度量变量是由双曲方程，而不强制执行的约束方程的演变。另一方面，虽然完全约束演化公式在理论上更有吸引力，并且应该导致更稳定和更准确的模拟，但这种方法并没有被广泛使用，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。多重网格方法求解微分方程具有一系列的离散化，其计算成本通常低于其他方法，如直接方法，松弛方法，逐次超松弛。使用多重网格加速，可以在与求解流体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经济实惠。作为第一步，以评估多重网格方法在相对论模拟的性能和鲁棒性，我们开发了一个流体力学代码，利用标准的有限体积法加上多重网格度量求解器来解决爱因斯坦方程的CFC近似。在本文中，我们提出的方法和实现我们的代码Gmunu和它的属性和性能在一些基准相对论流体力学问题。
英文摘要：
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.
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