中等质量比旋近的世界管切除方法：标量场玩具模型
Worldtube excision method for intermediate-mass-ratio inspirals: scalar-field toy model
AI总结：
本文针对中等质量比黑洞双星模拟中因尺度差异导致的计算成本问题，提出世界管切除方法，用含未知参数的解析模型替代小天体附近区域并通过匹配确定参数，并在标量场玩具模型中验证了不同匹配策略的可行性。
摘要：
黑洞双星旋近与并合模拟的计算成本与质量比 $q:=m_2/m_1\leq 1$ 的平方成反比增长。其中一个 $q$ 因子来自轨道周期数，它与 $1/q$ 成正比；另一个 $q$ 因子与每轨道所需的时间步数有关，该步数受 Courant-Friedrich-Lewy 条件约束，必须分辨两个差异巨大的长度尺度。这种不利的标度关系使得模拟在 $q$ 越小时越难以进行。本文提出并探索一种方法，用于缓解中等天体物理质量比范围（$10^{-4} \lesssim q\lesssim 10^{-2}$）内模拟中的尺度差异，在该范围内纯微扰方法可能并不充分。我们在数值区域中切除一个半径远大于 $m_2$ 的、围绕较小天体的区域，并用一个近似潮汐变形黑洞的解析模型来替代它。该解析模型包含某些先验未知的参数，这些参数与未知的物理细节以及规范调节项有关；它们通过与切除区域之外的数值解进行匹配而被动态确定。在本文中，我们发展了这一基本思想，并将其应用于一个玩具模型：一个标量电荷在史瓦西黑洞周围的圆测地轨道上运动，在 1+1 维框架下求解无质量 Klein-Gordon 场。我们的主要目标是探索不同匹配策略的效用和性质，为此我们开发了两个独立的实现：一个是有限差分实现，另一个是谱方法实现。我们讨论了将该方法推广到完整三维数值演化以及引力情形的途径。
英文摘要：
The computational cost of inspiral and merger simulations for black-hole binaries increases in inverse proportion to the square of the mass ratio $q:=m_2/m_1\leq 1$. One factor of $q$ comes from the number of orbital cycles, which is proportional to $1/q$, and another is associated with the required number of time steps per orbit, constrained (via the Courant-Friedrich-Lewy condition) by the need to resolve the two disparate length scales. This problematic scaling makes simulations progressively less tractable at smaller $q$. Here we propose and explore a method for alleviating the scale disparity in simulations with mass ratios in the intermediate astrophysical range ($10^{-4} \lesssim q\lesssim 10^{-2}$), where purely perturbative methods may not be adequate. A region of radius much larger than $m_2$ around the smaller object is excised from the numerical domain, and replaced with an analytical model approximating a tidally deformed black hole. The analytical model involves certain a priori unknown parameters, associated with unknown bits of physics together with gauge-adjustment terms; these are dynamically determined by matching to the numerical solution outside the excision region. In this paper we develop the basic idea and apply it to a toy model of a scalar charge in a circular geodesic orbit around a Schwarzschild black hole, solving for the massless Klein-Gordon field in a 1+1D framework. Our main goal here is to explore the utility and properties of different matching strategies, and to this end we develop two independent implementations, a finite-difference one and a spectral one. We discuss the extension of our method to a full 3D numerical evolution and to gravity.
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