#Mstar online code#
We note that regularized integration algorithms using particular tree structures have been implemented before: Jernigan & Porter ( 1989) developed a KS-regularized binary tree code while Mikkola & Aarseth ( 1989) proposed a ‘branching’ ks-chain structure for few-body regularization.
First, we use a minimum spanning tree (MST) coordinate system instead of the chain structure, motivating the name of the code. Our new code contains two main improvements compared to existing AR integrators. In this paper, we present a new algorithmically regularized (AR) integrator mstar developed and implemented with the aim of addressing some of the shortcomings of the previous algorithms. We note that similar performance issues with the ks-chain algorithm have already been reported in the literature see e.g. This is because the included ar-chain integrator becomes impractically inefficient with more than ∼300–500 particles in the regularized chain region. 2005) could easily run simulations with ∼10–100 times more stellar particles, up to N part ∼ 10 9 collisionless particles in a single galaxy. 2017) is currently limited to a particle resolution of N part ≲ 10 7 stellar particles per galaxy, although the underlying tree code gadget-3 (Springel et al. The regularized tree code KETJU (Rantala et al. 2017) use an implementation of the ar-chain integrator (Mikkola & Merritt 2006, 2008) to resolve the small-scale dynamics around SMBHs.ĭespite the many successes of regularization methods, their original design as few-body codes still limits their applicability to systems with a very large number of particles, which is crucial for performing galactic-scale simulations at high accuracy. 2008 Aarseth 2012) and regularized tree codes (Rantala et al. Many current direct summation codes (Harfst et al. This algorithmic regularization (AR) method is faster than the previous regularization methods and more accurate, especially in the case of large mass ratios between particles in a N-body system.
In the N-body codes of Aarseth ( 1999), the ks-chain treats the mutual encounters of not more than six particles simultaneously.Ī new regularization method, which does not require a coordinate transformation but only a time transformation, was discovered by Mikkola & Tanikawa ( 1999a) and Preto & Tremaine ( 1999). By arranging the simulation particles into a chain of inter-particle vectors, the ks-chain of Mikkola & Aarseth ( 1993) reduced the number of required KS transformations from N part( N part − 1)/2 to N part − 1, yielding a much more efficient regularization method. A major step forward for regularized dynamics was the introduction of the chain concept. The first such method with practical numerical applications was the two-body Kustaanheimo–Stiefel (KS) regularization method (Kustaanheimo & Stiefel 1965), which transformed both the time and spatial coordinates of the system. The key idea of regularization is to transform the equations of motion of a dynamical system into a form without coordinate singularities, which makes solving the dynamics significantly easier using standard numerical integration techniques (Aarseth 2003). The most widely used method to treat binaries and close encounters of simulation particles in the collisional regions of simulations is the technique of regularization. One of the major obstacles for developing such a code has been the lack of available fast accurate small-scale collisional post-Newtonian integrators, which are also straightforward to couple to both large-scale gravity solvers and hydrodynamical methods. We thus argue that a self-consistent numerical framework for simulating SMBH dynamics in a realistic collisional environment with a high stellar density and including also a gas component still remains to be developed. 2016), but this type of procedure is typically cumbersome and may introduce spurious numerical effects into the simulation. One possibility is to use on-the-fly code switching (Milosavljević & Merritt 2001 Khan et al. In addition, the most widely used direct summation codes are rarely coupled with a hydrodynamic solver. The twofold parallelization of the |$\mathcal ^2)$| scaling of the direct summation algorithm limits the applicability of this method to systems with much higher particle numbers. We present the novel algorithmically regularized integration method mstar for high-accuracy (|Δ E/E| ≳ 10 −14) integrations of N-body systems using minimum spanning tree coordinates.
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