Computational Relativistic Astrophysics via Space-Time Discontinuous Galerkin Finite Element Methods

计算相对论天体物理学领域已经进入了一个新的时代，它对引力波信号的预测与LIGO观测进行了比较，并被用于解释观测事件的特征。第一次观测到的引力波只有一年多一点的历史，已经使世界范围内对可能的引力波源进行建模的兴趣成倍增加。这个模型包括了其他类型的辐射，例如电磁或中微子辐射，在所谓的“多信使”天体物理学中。不久的将来将给我们带来突破性的发现，只能与X射线或射电频谱的发现带来的科学革命相比较。引力波天文学这一新领域的基础工作现在正在进行。 引力波是由由爱因斯坦方程支配的非常紧凑(密集)的天体发出的，例如涉及黑洞、中子星、双星系统的系统，或者形成黑洞或中子星的坍塌场景。对这样的系统进行忠实的建模不仅需要求解爱因斯坦方程，还需要对物质和辐射进行建模。这些系统是高度动态的，而详细、准确(忠实)的大规模数值计算是理解它们的唯一途径。控制方程太复杂了，无法用解析方法求解，也不能用简单的模型在台式计算机上计算。 在今天的高性能计算(HPC)系统上使用计算方法的复杂性和困难严重阻碍了进步。加速器(例如GPU)很常见，预计未来的系统将需要更多的并行性，同时提供更少的内存带宽，从而增加科学程序员的负担。随着硬件体系结构的发展，许多以前高效的算法已经不再有效，因为它们只利用了较新硬件计算能力的一小部分。 在本文提出的工作中，我们将开发新的数值算法来解决这些问题。虽然广义相对论以一种非常优雅的形式将时空视为一个单一的结构，但当前的主流数值方法并非如此：它们明确地将时空分割为空间和时间，以获得为非相对论场景设计的大量数值方法，但在这个过程中也放弃了相对论的许多优雅之处。 这里开发的数值方法将离散时空，而不是分开空间和时间，通过这样做，奇怪的是，将有可能成为一个更可伸缩和更有效的数量级。这反过来将允许模型在物理上更准确和更逼真，因为更多的物理可以被纳入到它们的描述中。

The field of Computational Relativistic Astrophysics has entered a new era where its predictions of gravitational wave signals are compared to LIGO observations, and are used to interpret signatures of observed events. The first observed gravitational wave is only a little more than one year old, and has already multiplied the world-wide interest in modelling possible gravitational wave sources. This modelling includes other kinds of radiation, such as electromagnetic or neutrino counterparts, in so-called "multi-messenger" astrophysics. The very near future will bring us breakthrough discoveries, to be compared only to the scientific revolution brought about by the discovery of X-rays or the radio spectrum. The groundwork in the new field of gravitational wave astronomy is being laid now. Gravitational waves are emitted by very compact (dense) astrophysical objects which are governed by the Einstein equations, such as systems involving black holes, neutron stars, binary systems of these, or collapse scenarios where black holes or neutron stars are formed. Faithfully modelling such systems requires not only solving the Einstein equations, but also modelling matter and radiation. These systems are highly dynamic, and detailed, accurate (faithful) large-scale numerical calculations are the only road towards understanding them. The governing equations are far too complex to be solved analytically, or in simple models that can be calculated on a desktop computer. Progress is severely hindered by the complexity and difficulties in using computational methods on today's high-performance computing (HPC) systems. Accelerators (e.g. GPUs) are commonplace, and future systems are expected to require even more parallelism while providing less memory bandwidth, increasing the burden of scientific programmers. As hardware architectures evolve, many formerly highly efficient algorithms are not efficient any more, as they only make use of a small fraction of the computing power of newer hardware. In the work proposed here, we will develop novel numerical algorithms to address these issues. While general relativity treats spacetime as a single construct in a very elegant formulation, current mainstream numerical methods do not: They explicitly split spacetime into space and time to gain access to large body of numerical methods designed for non-relativistic scenarios, but also foregoing much of the elegance of relativity in the process. The numerical methods developed here will discretize spacetime, not space and time separately, and by doing so, will curiously have the potential to be an order of magnitude more scalable and efficient. This will in turn allow models that are significantly more physically accurate and realistic, as more physics can be incorporated into their description.