Optimized Forest-Ruth Algorithm In Extended Phase Space And Its Application

The fourth-order explicit Forest-Ruth symplectic algorithm is constructed for a separable Hamiltonian system.It can be optimized by minimizing the sum of squares of the fifth-order truncated error term and obtain the corresponding optimization coefficients and optimization algorithms.Since explicit symplectic algorithm can not be directly applied for inseparable Hamiltonian systems,the traditional method is to use implicit symplectic algorithm to solve inseparable system.Whereas,due to the fact that implicit symplectic algorithm requires multiple iterations to achieve predetermined accuracy,it may increases the time-consumption and reduces the calculating efficiency,therefore applying explicit algorithm is still an important approach to considered.With the development of Pihajoki’s extended phase space method,it becomes possible to the apply traditional explicit symplectic algorithms in inseparable systems.By doubling the phase space variables,two Hamiltonian can be constructed in expression of original coordinates,extended momentum and the original momentum,extended coordinates,respectively,and all can be solved by explicit symplectic algorithm.In order to ensure that the original phase space and the extended phase space are equal in the integration,it is necessary to apply a permutation between the two sets of coordinate and momentum in order to control the error between the two sets of Hamiltonian quantities.The algorithm can be symplectic in the extended phase space,but the symplecticity can’t be guaranteed when mapped to the original space.However,it is a symmetric method with energy-saving properties like the symplectic algorithm.Then,the optimized Forest-Ruth algorithm combined with extended phase space method and permutation can construct the extended phase space optimized Forest-Ruth algorithm for inseparable systems.Numerical experiments show that:Firstly,the optimization algorithm can significantly improve the accuracy over the non-optimized algorithm in systems such as the planar circular restricted three-body problem and the non-spinning compact binary system;Secondly,the discussion of numerical difference brought by different permutation methods,such as series of coordinate and momentum permutation and the midpoint permutation,indicates that the midpoint permutation can make the optimized Forest-Ruth algorithm have better numerical performance.