A new family of high order one-step symplectic integration schemes for separable Hamiltonian systems with Hamiltonians of the form
T
(
p
)
+
U
(
q
)
is presented. The new integration methods are ...defined in terms of an explicitly defined generating function (of the third kind). They are implicit in
q
(but explicit in
p
and the internal states), and require the evaluation of the gradients of
T
(
p
) and
U
(
q
) and the actions of their Hessians on vectors (the later being relatively cheap in the case of many-body problems). A time-symmetric symplectic method is constructed that has order 10 when applied to Hamiltonian systems with quadratic kinetic energy
T
(
p
). It is shown by numerical experiments that the new methods have the expected order of convergence.
High precision symplectic integrators for the Solar System Farrés, Ariadna; Laskar, Jacques; Blanes, Sergio ...
Celestial mechanics & dynamical astronomy/Celestial mechanics and dynamical astronomy,
06/2013, Letnik:
116, Številka:
2
Journal Article
Recenzirano
Odprti dostop
Using a Newtonian model of the Solar System with all 8 planets, we perform extensive tests on various symplectic integrators of high orders, searching for the best splitting scheme for long term ...studies in the Solar System. These comparisons are made in Jacobi and heliocentric coordinates and the implementation of the algorithms is fully detailed for practical use. We conclude that high order integrators should be privileged, with a preference for the new
method of Blanes et al. (
2013
).
Efficient computation of the Zassenhaus formula Casas, Fernando; Murua, Ander; Nadinic, Mladen
Computer physics communications,
November 2012, 2012-11-00, 20121101, Letnik:
183, Številka:
11
Journal Article
Recenzirano
Odprti dostop
A new recursive procedure to compute the Zassenhaus formula up to high order is presented, providing each exponent in the factorization directly as a linear combination of independent commutators and ...thus containing the minimum number of terms. The recursion can be easily implemented in a symbolic algebra package and requires much less computational effort, both in time and memory resources, than previous algorithms. In addition, by bounding appropriately each term in the recursion, it is possible to get a larger convergence domain of the Zassenhaus formula when it is formulated in a Banach algebra.
Majorant series for the N-body problem Antoñana, Mikel; Chartier, Philippe; Murua, Ander
International journal of computer mathematics,
01/2022, Letnik:
99, Številka:
1
Journal Article
Recenzirano
As a follow-up of a previous work of the authors, this work considers uniform global time-renormalization functions for the gravitational
N-body problem. It improves on the estimates of the radii of ...convergence obtained therein by using a completely different technique, both for the solution to the original equations and for the solution of the renormalized ones. The aforementioned technique which the new estimates are built upon is known as majorants and allows for an easy application of simple operations on power series. The new radii of convergence so-obtained are approximately doubled with respect to our previous estimates. In addition, we show that majorants may also be constructed to estimate the local error of the implicit midpoint rule (and similarly for Runge-Kutta methods) when applied to the time-renormalized N-body equations and illustrate the interest of our results for numerical simulations of the Solar System.
We present a practical algorithm based on symplectic splitting methods intended for the numerical integration in time of the Schrödinger equation when the Hamiltonian operator is either ...time-independent or changes slowly with time. In the later case, the evolution operator can be effectively approximated in a step-by-step manner: first divide the time integration interval in sufficiently short subintervals, and then successively solve a Schrödinger equation with a different time-independent Hamiltonian operator in each of these subintervals. When discretized in space, the Schrödinger equation with the time-independent Hamiltonian operator obtained for each time subinterval can be recast as a classical linear autonomous Hamiltonian system corresponding to a system of coupled harmonic oscillators. The particular structure of this linear system allows us to construct a set of highly efficient schemes optimized for different precision requirements and time intervals. Sharp local error bounds are obtained for the solution of the linear autonomous Hamiltonian system considered in each time subinterval. Our schemes can be considered, in this setting, as polynomial approximations to the matrix exponential in a similar way as methods based on Chebyshev and Taylor polynomials. The theoretical analysis, supported by numerical experiments performed for different time-independent Hamiltonians, indicates that the new methods are more efficient than schemes based on Chebyshev polynomials for all tolerances and time interval lengths. The algorithm we present automatically selects, for each time subinterval, the most efficient splitting scheme (among several new optimized splitting methods) for a prescribed error tolerance and given estimates of the upper and lower bounds of the eigenvalues of the discretized version of the Hamiltonian operator.
We propose an implementation of symplectic implicit Runge-Kutta schemes for highly accurate numerical integration of non-stiff Hamiltonian systems based on fixed point iteration. Provided that the ...computations are done in a given floating point arithmetic, the precision of the results is limited by round-off error propagation. We claim that our implementation with fixed point iteration is near-optimal with respect to round-off error propagation under the assumption that the function that evaluates the right-hand side of the differential equations is implemented with machine numbers (of the prescribed floating point arithmetic) as input and output. In addition, we present a simple procedure to estimate the round-off error propagation by means of a slightly less precise second numerical integration. Some numerical experiments are reported to illustrate the round-off error propagation properties of the proposed implementation.
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