A time-symmetric individual block time step algorithm in n-body integration

Ayrık blok zaman adımlı integrasyon algoritmaları, çok-cisim problemlerinde sıklıkla kullanılan algoritmalardandır. Farklı zaman ölçeklerine sahip cisimlerin aynı problemin parçaları olduğu durumlarda, kullanımları kaçınılmazdır. Ancak kullanılan integrasyon şeması zaman-simetrik bile olsa, blok zaman adımları kullanıldığında zaman simetrisi bozulmaktadır. Zaman simetrisinin bozulmasının doğal sonucu, toplam enerji üzerindeki hata birikiminin doğrusal olarak artmasıdır. Bu çalışmada, zaman-simetrik blok zaman adımlı bir algoritma geliştirilmiştir. Geliştirilen algoritma, sabit zaman adımları kullanıldığında zaman-simetrik yapısını koruyan integrasyon şemalarının, değişken ayrık blok adımlı bir algoritma içerisinde de zaman simetrik yapısını korumasını sağlamaktadır. Hatta integrasyon şeması tam olarak zaman simetrik olmasa bile, geliştirilen algoritma ile yüksek enerji korunumu sağlandığı gözlenmiştir. Özellikle uzun süreli integrasyonlarda algoritmanın yüksek başarımı belirgindir. Zaman-simetrik ayrık blok zaman adımlı algoritma, bir yineleme yapısı içermektedir. Bu yinelemenin ayrık zaman adımlı integrasyon şemasıyla uygulanabilmesi için, çağ aralıkları (era) belirlenmiş, yineleme işlemleri bu çağ aralıkları üzerinden sağlanmıştır. Yapılan testlerde bu çağ aralıklarının büyüklüğünün hem yineleme sayısı, hem de bellek tüketimi açısından önemli olduğu gözlenmiştir. Çalışmanın bu safhasında, algoritma çağ aralıklarının değişken tutulabildiği bir yapıya kavuşturulmuştur. Algoritma Leapfrog integrasyon şeması kullanılarak geliştirilmiş, etkinliği daha yüksek dereceden integrasyon şemalarıyla da sınanmıştır. İçerdiği yineleme yapısı yüzünden ayrık zaman adımlı önceki algoritmalardan daha yüksek zaman ve bellek tüketimine sahiptir. Ancak toplam enerji üzerindeki hata birikimlerini büyük ölçüde azaltmaktadır. Anahtar Kelimeler: Çok-cisim problemi, sayısal integrasyon, ayrık blok-zaman adımları.The gravitational N-body problem can not be solved analytically except for two-body cases. While there are various approximate methods, such as Monte Carlo or Fokker-Planck method, the most accurate way to solve the N-body problem is by direct integration of the orbits of the N bodies. Direct integration of the N-body problem is a necessity for many cases such as star cluster simulations. During the last fifteen years, two approaches have been put forward to improve numerical conservation of energy and other theoretically conserved quantities: symplectic integration schemes, where the simulated system is guaranteed to follow a slightly perturbed Hamiltonian system, and time-symmetric integration schemes, where the simulated system follows the same trajectory in phase space, when run backward or forward. Time-symmetric integration schemes share with symplectic schemes the property that their energy errors show a much better behavior than the case for generic integration schemes. Allowing adaptive time steps typically leads to a loss of symplecticity. In contrast, time symmetry can be easily maintained, at least for a continuous choice of time step size. An iterative scheme must be combined with individual block time step scheme to apply the new algorithm to the n-body problem effectively. However, straightforward implementation of time-symmetry, translated to block time steps, faces significant hurdles for N-body problem. For example, iteration can lead to oscillatory behavior, and even when such behavior is suppressed, energy errors show a linear drift in time. In this work it is aimed to construct a time symmetric block time step integration scheme. We first consider to analyze block time step scheme with time symmetrization procedure for the gravitational two-body problem. It is hard to generate an efficient algorithm for N -body problem without deep and clear understanding of this fundamental problem. However it is easy to correct the algorithm in Kepler problem. Besides, it is known that contributions to the energy error are largely generated by close encounters between two particles. For these reasons, two-body problem is preferred to develop the algorithm in the first instance. The algorithm is developed and tested in two-body problem for accuracy and energy conservations in many different Kepler problem with different iteration numbers. In the process of the development work, leapfrog integration scheme which is the second order time-symmetry scheme is preferred. Also fourth, sixth, and eighth order Hermite integration schemes are used for test runs. The major conceptual difficulty in designing a time-symmetric block time step scheme is the global context information that is needed, with extensions toward the future as well as the past. We present an approach that circumvents these problems. We divide and conquer: we split the total history of our simulation into a number of smaller periods, which we call every one as era. Each era extends a period in time equal to the largest allowed time step tmax, or to an integer multiple of tmax, whatever turns out to be the most convenient. For such reasons, we have analyzed the era concept in more detail for time symmetrized block time steps. Our test results show that size of era must be chosen carefully for stable and robust integrations. This is important especially for long term simulations with highly desirable energy conservations. Era size is also important to avoid extra data storage and uselessly high number of iterations which require too much running times. As a second gain of the work, we re-designed the previous scheme, and suggested dynamically changing size for era. With this scheme, iteration process of the integration can follow needs of the simulation adaptively. And, era size will be well-adjusted with physics of the problem with proper criterion. Our basic integrator in the work is leapfrog scheme. It is a well known, second order, time symmetric integration scheme for fixed time steps. We also used fourth, sixth, and eighth order Hermite schemes to see the ability of the algorithm with higher order integration methods. We used different Kepler problems in two-body tests, and used Plummer model initial conditions for N-body tests. The algorithm does not show linear growing error with time symmetric integrations. It also shows high energy conservations with other integration schemes especially for long integration times. Keywords: N-Body integration, time symmetry, individual block - time step, Hermite integration