Numerical relativity in spherical polar coordinates: Evolution calculations with the BSSN formulation
Numerical relativity in spherical polar coordinates: Evolution calculations with the BSSN formulation
复制标题
球极坐标中的数值相对论：使用 BSSN 公式进行演化计算
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
E. Muller
中科院分区：
文献类型：
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作者：
T. Baumgarte;P. J. Montero;I. Cordero;E. Muller
在缺乏对称性假设的情况下，大多数数值相对论模拟采用笛卡尔坐标。虽然笛卡尔坐标具有一些理想的特性，但球极坐标似乎更适合某些应用，包括引力坍缩和超新星模拟。由于需要处理原点和轴上的坐标奇点，例如通过对适当变量进行仔细正则化，球极坐标中数值相对论代码的开发受到了阻碍。假设球对称并采用 Baumgarte-Shapiro-Shibata-Nakamura 方程的协变版本，Montero 和 Cordero-Carri\'on 最近证明，当部分隐式 Runge-Kutta 方法用于引力场的时间演化时，这种正则化是不必要的。在这里，我们报告了 Baumgarte-Shapiro-Shibata-Nakamura 方程在球极坐标中的实现，没有任何对称性假设。使用部分隐式龙格-库塔方法，我们获得了三个空间维度的稳定模拟，而无需正则化原点或轴。我们进行并讨论了许多测试来评估代码的稳定性、准确性和收敛性，即弱引力波、平衡状态下球形和旋转相对论恒星的“水-无水”演化以及单个黑洞。
In the absence of symmetry assumptions most numerical relativity simulations adopt Cartesian coordinates. While Cartesian coordinates have some desirable properties, spherical polar coordinates appear better suited for certain applications, including gravitational collapse and supernova simulations. Development of numerical relativity codes in spherical polar coordinates has been hampered by the need to handle the coordinate singularities at the origin and on the axis, for example by careful regularization of the appropriate variables. Assuming spherical symmetry and adopting a covariant version of the Baumgarte-Shapiro-Shibata-Nakamura equations, Montero and Cordero-Carri\'on recently demonstrated that such a regularization is not necessary when a partially implicit Runge-Kutta method is used for the time evolution of the gravitational fields. Here we report on an implementation of the Baumgarte-Shapiro-Shibata-Nakamura equations in spherical polar coordinates without any symmetry assumptions. Using a partially implicit Runge-Kutta method we obtain stable simulations in three spatial dimensions without the need to regularize the origin or the axis. We perform and discuss a number of tests to assess the stability, accuracy and convergence of the code, namely weak gravitational waves, ``hydro-without-hydro'' evolutions of spherical and rotating relativistic stars in equilibrium, and single black holes.