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Efficiency of Runge-Kutta Methods in Solving Kepler Problem

机译：runge-kutta方法在解决开普勒问题中的效率

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The aim of this research is to study the efficiency of symplectic and non-symplectic Runge-Kutta methods in solving Kepler problem. The numerical behavior of the Runge-Kutta (RK) methods that are symmetric such as the implicit midpoint rule (IMR), implicit trapezoidal rule (ITR), 2-stage and 2-stage Gauss (G2) method are compared with the non-symmetric Runge-Kutta methods such as the explicit and implicit Euler (EE and IE), explicit midpoint rule (EIMR), explicit trapezoidal rules (EITR), explicit 4-stage Runge-Kutta (RK4) method and 2-stage Radau IIA method (R2A). Kepler problem is one type of nonlinear Hamiltonian problem that describes the motion in a plane of a material point that is attracted towards the origin with a force inversely proportional to the distance squared. The exact solutions phase diagram produces a unit circle. The non-symplectic methods only reproduce a unit circle at certain time intervals while the symplectic methods do produce a unit circle at any time intervals. Some phase diagram show spiral in or spiral out patterns which means the solutions are running away from the unit circle. This also means that the absolute error will be increasing in long time integration. The numerical experiments for the Kepler problem are given for many time intervals and the results show that the most efficient method is G2 of order-4 and surprisingly RK4 seems to be efficient too although it is not a symplectic nor a symmetric method. The numerical results on Kepler problem concluded that, the higher the order of the method, the most efficient the method can be in solving Kepler problem despite whether they are explicit or implicit or symmetric and symplectic.
机译：本研究的目的是研究互相兼并的旋转速率 - 库特拉方法的效率在解决开普勒问题方面。与非隐式中点规则（IMR），隐式梯形规则（ITR），2级和2级高斯（G2）方法进行对称的runge-Kutta（RK）方法的数值行为与非 - 非诸如显式和隐式欧拉（EE和IE），显式中点规则（EIMR），显式梯形规则（EITR），显式4级跑步 - 库特拉（RK4）方法和2阶段Radau IIA方法的对称和隐式欧拉（EIMR）。 （R2A）。开普勒问题是一种非线性汉密尔顿问题的一种非线性哈密尔顿问题，其描述了一种物料点的平面中的运动，其朝向原点吸引，其力与距离平方成正比成反比。确切的解决方案相图产生单位圆圈。非杂项方法仅在特定时间间隔再现单位圆，而杂项方法确实以任何时间间隔产生单位圆。一些相图显示了螺旋形或螺旋形图案，这意味着解决方案远离单位圆圈。这也意味着在长时间集成中绝对误差将增加。对于许多时间间隔给出了开普勒问题的数值实验，结果表明，最有效的方法是订单-4的G2，令人惊讶的RK4似乎也很有效，尽管它不是一个辛也是对称的方法。关键词问题的数值结果得出结论，该方法的顺序越高，尽管它们是明确的或隐含的还是对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的或对称的，但方法的数值结果得出结论所知，最有效的方法可以在解决开普勒问题。

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