The computational cost of inspiral and merger simulations for black hole binaries increases in inverse proportion to the square of the mass ratio $q≔m_2/m_1\le 1$. 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-Friedrichs-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 them 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 $(1+1)\mathrm{D}$ 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.

中文翻译：

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中等质量比螺旋体的 Worldtube 切除方法：标量场玩具模型

黑洞双星的内旋和合并模拟的计算成本与质量比的平方成反比增加 $q≔{米}_2/{米}_1\le 1$. 因素之一$q$ 来自轨道周期数，它与 $1/q$，另一个与每个轨道所需的时间步数相关联，受解决两个不同长度尺度的需要的约束（通过 Courant-Friedrichs-Lewy 条件）。这种有问题的缩放使模拟越来越难以处理$q$. 在这里，我们提出并探索了一种减轻质量比在中间天体物理范围内的模拟中的尺度差异的方法（${10}^{-4}\lesssim q\lesssim {10}^{-2}$)，在这种情况下，纯微扰方法可能不够用。半径远大于${米}_2$从数值域中切除较小物体周围的物体，取而代之的是一个近似潮汐变形黑洞的分析模型。分析模型涉及某些先验未知参数，与物理的未知位以及规范调整项相关；这些是通过将它们与切除区域外的数值解进行匹配来动态确定的。在本文中，我们开发了基本思想并将其应用于围绕施瓦西黑洞的圆形测地线轨道中的标量电荷的玩具模型，求解无质量的 Klein-Gordon 场$(1+1)\mathrm{D}$框架。我们在这里的主要目标是探索不同匹配策略的效用和特性，为此我们开发了两种独立的实现，一种是有限差分的，一种是光谱的。我们讨论将我们的方法扩展到完整的 3D 数值演化和重力。