Word series for dynamical systems and their numerical integrators

We study word series and extended word series, classes of formal series for the analysis of some dynamical systems and their discretizations. These series are similar to but more compact than B-series. They may be composed among themselves by means of a simple rule. While word series have appeared before in the literature, extended word series are introduced in this paper. We exemplify the use of extended word series by studying the reduction to normal form and averaging of some perturbed integrable problems. We also provide a detailed analysis of the behavior of splitting numerical methods for those problems.A. Murua and J.M. Sanz-Serna have been supported by Projects MTM2013-46553-C3-2-P and MTM2013-46553-C3-1-P from Ministerio de Economía y Comercio, Spain. Additionally A. Murua has been partially supported by the Basque Government (Consolidated Research Group IT649-13)

An implicit symplectic solver for high-precision long term integrations of the Solar System

We present FCIRK16, a 16th-order implicit symplectic integrator for long-term high precision Solar System simulations. Our integrator takes advantage of the near-Keplerian motion of the planets around the Sun by alternating Keplerian motions with corrections accounting for the planetary interactions. Compared to other symplectic integrators (the Wisdom and Holman map and its higher order generalizations) that also take advantage of the hierarchical nature of the motion of the planets around the central star, our methods require solving implicit equations at each time-step. We claim that, despite this disadvantage, FCIRK16 is more efficient than explicit symplectic integrators for high precision simulations thanks to: (i) its high order of precision, (ii) its easy parallelization, and (iii) its efficient mixed-precision implementation which reduces the effect of round-off errors. In addition, unlike typical explicit symplectic integrators for near Keplerian problems, FCIRK16 is able to integrate problems with arbitrary perturbations (non necessarily split as a sum of integrable parts). We present a novel analysis of the effect of close encounters in the leading term of the local discretization errors of our integrator. Based on that analysis, a mechanism to detect and refine integration steps that involve close encounters is incorporated in our code. That mechanism allows FCIRK16 to accurately resolve close encounters of arbitrary bodies. We illustrate our treatment of close encounters with the application of FCIRK16 to a point mass Newtonian 15-body model of the Solar System (with the Sun, the eight planets, Pluto, and five main asteroids) and a 16-body model treating the Moon as a separate body. We also present some numerical comparisons of FCIRK16 with a state-of-the-art high order explicit symplectic scheme for 16-body model that demonstrate the superiority of our integrator when very high precision is required.Consolidated Research Group MATHMODE (IT1294-19

An implicit symplectic solver for high-precision long term integrations of the Solar System

Compared to other symplectic integrators (the Wisdom and Holman map and its higher order generalizations) that also take advantage of the hierarchical nature of the motion of the planets around the central star, our methods require solving implicit equations at each time-step. We claim that, despite this disadvantage, FCIRK16 is more efficient than explicit symplectic integrators for high precision simulations thanks to: (i) its high order of precision, (ii) its easy parallelization, and (iii) its efficient mixed-precision implementation which reduces the effect of round-off errors. In addition, unlike typical explicit symplectic integrators for near Keplerian problems, FCIRK16 is able to integrate problems with arbitrary perturbations (non necessarily split as a sum of integrable parts). We present a novel analysis of the effect of close encounters in the leading term of the local discretization errors of our integrator. Based on that analysis, a mechanism to detect and refine integration steps that involve close encounters is incorporated in our code. That mechanism allows FCIRK16 to accurately resolve close encounters of arbitrary bodies. We illustrate our treatment of close encounters with the application of FCIRK16 to a point mass Newtonian 15-body model of the Solar System (with the Sun, the eight planets, Pluto, and five main asteroids) and a 16-body model treating the Moon as a separate body. We also present some numerical comparisons of FCIRK16 with a state-of-the-art high order explicit symplectic scheme for 16-body model that demonstrate the superiority of our integrator when very high precision is required

Higher-order averaging, formal series and numerical integration III: error bounds

In earlier papers, it has been shown how formal series like those used nowadays to investigate the properties of numerical integrators may be used to construct high-order averaged systems or formal first integrals of Hamiltonian problems. With the new approach the averaged system (or the formal first integral) may be written down immediately in terms of (i) suitable basis functions and (ii) scalar coefficients that are computed via simple recursions. Here we show how the coefficients/basis functions approach may be used advantageously to derive exponentially small error bounds for averaged systems and approximate first integrals.A. Murua and J.M. Sanz-Serna have been supported by projects MTM2010-18246-C03-03 and MTM2010-18246-C03-01 respectively from Ministerio de Ciencia e Innovación.Publicad

An Intrinsic Description of the Nonlinear Aeroelasticity of Very Flexible Wings

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/90662/1/AIAA-2011-1917-972.pd

Backward error analysis and the substitution law for Lie group integrators

Butcher series are combinatorial devices used in the study of numerical methods for differential equations evolving on vector spaces. More precisely, they are formal series developments of differential operators indexed over rooted trees, and can be used to represent a large class of numerical methods. The theory of backward error analysis for differential equations has a particularly nice description when applied to methods represented by Butcher series. For the study of differential equations evolving on more general manifolds, a generalization of Butcher series has been introduced, called Lie--Butcher series. This paper presents the theory of backward error analysis for methods based on Lie--Butcher series.Comment: Minor corrections and additions. Final versio

Body composition in breast cancer survivors in Sonorenses women

INTRODUCTION: Breast cancer is the most common type of cancer in the worldwide. In the same sense, this disease is one of the most common cancers affecting Mexican women. In the year 2014 in México, there were 11,372 new cases of breast cancer with an incidence rate of 22.56 per 100,000 in habitants older than 10 years. Women with breast cancer are often subjected to an operation due to this affectation which decreases its functionality and body composition. PURPOSE: To examine the body composition in breast cancer survivors in a sample of women from Hermosillo, Sonora, México. METHODS: This study was a cross-sectional descriptive study design. 21 women with breast cancer who had been operated on left arm and had been recruited at one university-based exercise program for breast cancer survivors in Hermosillo, Sonora, México. Body composition (BC) was measured. The right arm non-operated was considered as control. Statistical difference between the operated versus non-operated arm were determined with t-student test for independent samples. RESULTS: In the present study, body fat (1719.1 ± 456.7 vs. 1819.8 ± 467.9 grams, p ≤ 0.05), lean mass (1960.2 ± 308.7 vs. 2151.5 ± 313.5 grams, p ≤ 0.05) and total body mass (3679.3 ± 643.3 vs. 3971.1 ± 675.9 grams, p ≤ 0.05) of the left operated arm of women who breast cancer survivors were significantly lower than the mean of the right non-operated arm. CONCLUSION: Breast cancer survivors’ women who have underwent an operation on their left arm sowed a lower percentage of fat, fat mass and total mass compared to their non-operate arm. The present study underline the importance to apply rehabilitation or exercise program focused to reduce the changes in the body composition