Computational Methods for Astrophysical Flows

天体物理流体动力学是物理学的一个分支，涉及理解遥远物体（如黑洞和中子星）的演化。为了充分理解这些物体，数学模型必须包含广义相对论、电磁学和流体动力学效应。得到的方程是一个大的、耦合的、非线性的偏微分方程组，其中一些是演化方程，而另一些是由各种规范自由度产生的约束方程。PI的研究将集中于开发非结构化网格的高阶方案，以解决各种简化版本的完整天体物理流体动力学模型。例如，天体物理学家非常感兴趣的一个问题是黑洞的质量吸积和由此形成的相对论性喷流；这种现象可以在测试流体极限（即背景时空度量是固定的）中处理。另一个重要的问题是两个大质量黑洞碰撞产生的引力波（即时空涟漪）；这个问题可以首先从最小耦合标量场的极限来考虑。PI将利用不连续伽辽金格式和残差分布格式两种方法来构造准确有效的格式。特别是，这些方法将与自适应网格细化策略相结合。为了有效地做到这一点，PI将构建一个后验误差估计器，该估计器可用于动态诊断产生较大数值误差的地方。所得的一套数值方法将被编入计算机代码，并在网上免费提供。虽然天体物理对象，如黑洞吸积盘、河外喷流和超新星可以用各种望远镜观测到，但直接实验显然是不可能的。另一方面，试图解释这些物体的物理性质的数学模型必然是复杂的，必须包括引力、电磁和流体动力学效应。得到的数学方程的精确解只能在非常特殊的情况下才能得到。因此，从科学的角度理解天体物理现象的能力在很大程度上取决于运行准确和有效的计算机模拟的能力，而这反过来又取决于用于执行这些模拟的计算方法的质量。PI的研究重点是开发求解天体物理流体动力学方程的高阶计算方法。这项工作的一个方面将涉及各种误差指标的构建，这些指标可用于动态诊断和纠正计算的准确性。另一个方面将是开发一个软件包，将在网上免费提供。由这一发展产生的计算方法将应用于天体物理学中的两个不同问题：(1)黑洞吸积过程中天体物理喷流的形成；(2)两个黑洞相互作用的动力学以及由此产生的引力波。

Astrophysical fluid dynamics is a branch of physics concerned with understanding the evolution of far-away objects such as black holes and neutron stars. In order to fully understand such objects, mathematical models must incorporate general relativistic, electromagnetic, and fluid dynamic effects. The resulting equations are a large, coupled, nonlinear system of partial differential equations, some of which are evolution equations, while others are constraint equations that result from various gauge freedoms. The PI's research will focus on developing high-order schemes on unstructured grids to solve various simplified versions of the full astrophysical fluid dynamic model. For example, one problem of great interest to astrophysicists is that of mass accretion onto black holes and the resulting formation of relativistic jets; this phenomenon can be treated in the test-fluid limit (i.e., background spacetime metric is fixed). Another important problem is the generation of gravitational waves (i.e., ripples in spacetime) from the collision of two massive black holes; this problem can be first looked at in the minimally coupled scalar field limit. The PI will make use of both discontinuous Galerkin and residual distribution scheme methodologies to construct accurate and efficient schemes. In particular, these methods will be combined with adaptive mesh refinement strategies. In order to do this efficiently, the PI will construct a posteriori error estimators that can be used to dynamically diagnose where large numerical errors are being made. The resulting set of numerical methods will be incorporated into a computer code that will be made freely available on the web.Although astrophysical objects such as black hole accretion disks, extragalactic jets, and supernovae are observable using various telescopes, direct experimentation is clearly not possible. On the other hand, mathematical models that attempt to explain the physics of these objects are necessarily complex and must include gravitational, electromagnetic, and fluid dynamic effects. Exact solutions to the resulting mathematical equations can only be constructed in very special cases. Therefore, the ability to understand astrophysical phenomena from a scientific viewpoint rests largely on the ability to run accurate and efficient computer simulations, which, in turn, rests on the quality of the computational methods that are used to carry out those simulations. The PI's research is focused on developing classes of high-order computational methods for solving the equations of astrophysical fluid dynamics. One aspect of this work will involve the construction of various error indicators that can be used to dynamically diagnose and correct the accuracy of a computation. Another aspect will be to develop a software package that will be made freely available on the web. The computational methods that result from this development will be applied to two distinct problems in astrophysics: (1) the formation of astrophysical jets from black hole accretion processes and (2) the dynamics of the interaction of two black holes and the resulting generation of gravitational waves.