Combining spectral and shock-capturing methods: A new numerical approach for 3D relativistic core collapse simulations
Combining spectral and shock-capturing methods: A new numerical approach for 3D relativistic core collapse simulations
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结合光谱和冲击捕获方法：3D 相对论性核心塌陷模拟的新数值方法
DOI：
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发表时间：
2004
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通讯作者：
E. Mueller
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文献类型：
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作者：
H. Dimmelmeier;J. Novak;J. Font;J. Ibáñez;E. Mueller
我们提出了一个新的三维广义相对论流体动力学程序，该程序用于模拟恒星核坍缩成中子星星，以及旋转相对论恒星的脉动和不稳定性。与大多数现有的基于笛卡尔坐标的三维数值相对论代码中遵循的常见方法相反，在该代码中，度量和流体动力学方程都使用球极坐标来制定和数值求解。这个新代码的一个显着特点是两种类型的精确数值方案的组合，专门设计来解决每个方程组。更准确地说，该代码使用谱方法来求解引力场方程，这些方程是在三度规的共形平坦条件（CFC）假设下制定的。相应地，流体动力学方程求解一类有限差分方法称为高分辨率激波捕获计划，基于国家的最先进的黎曼求解器和三阶细胞重建程序。我们证明了有限差分网格和谱网格的组合，流体力学和度量方程，分别解决，可以成功地完成。这种方法，我们称之为Mariage des Maillages（法语网格婚礼），导致度量求解器的高精度，并且在实践中，更多»允许使用计算负担得起的资源进行完全三维的应用，沿着确保演化的长期数值稳定性。我们比较我们的新方法，其他两个，有限差分为基础的方法来解决度量方程，我们已经在早期的轴对称模拟核心崩溃。在二维和三维的各种测试，涉及高度扰动的中子星星时空和（轴对称）恒星核心崩溃，这表明有能力的代码处理时空的对称性和不对称性在强引力。这些测试也被用来评估重力波形提取能力的代码，这是基于牛顿四极公式。这里提出的代码不仅限于近似的爱因斯坦方程，如CFC，但它也很适合，在原则上，最近的约束配方的度量方程，椭圆方程有一个卓越的双曲方程。«少
We present a new three-dimensional general relativistic hydrodynamics code which is intended for simulations of stellar core collapse to a neutron star, as well as pulsations and instabilities of rotating relativistic stars. Contrary to the common approach followed in most existing three-dimensional numerical relativity codes which are based in Cartesian coordinates, in this code both the metric and the hydrodynamics equations are formulated and solved numerically using spherical polar coordinates. A distinctive feature of this new code is the combination of two types of accurate numerical schemes specifically designed to solve each system of equations. More precisely, the code uses spectral methods for solving the gravitational field equations, which are formulated under the assumption of the conformal flatness condition (CFC) for the three-metric. Correspondingly, the hydrodynamics equations are solved by a class of finite difference methods called high-resolution shock-capturing schemes, based upon state-of-the-art Riemann solvers and third-order cell-reconstruction procedures. We demonstrate that the combination of a finite difference grid and a spectral grid, on which the hydrodynamics and metric equations are, respectively, solved, can be successfully accomplished. This approach, which we call Mariage des Maillages (French for grid wedding), results in high accuracy of the metric solver and, in practice,more » allows for fully three-dimensional applications using computationally affordable resources, along with ensuring long-term numerical stability of the evolution. We compare our new approach to two other, finite difference based, methods to solve the metric equations which we already employed in earlier axisymmetric simulations of core collapse. A variety of tests in two and three dimensions is presented, involving highly perturbed neutron star spacetimes and (axisymmetric) stellar core collapse, which demonstrate the ability of the code to handle spacetimes with and without symmetries in strong gravity. These tests are also employed to assess the gravitational waveform extraction capabilities of the code, which is based on the Newtonian quadrupole formula. The code presented here is not limited to approximations of the Einstein equations such as CFC, but it is also well suited, in principle, to recent constrained formulations of the metric equations where elliptic equations have a preeminence over hyperbolic equations.« less