The Experts below are selected from a list of 117 Experts worldwide ranked by ideXlab platform
Ari Silburt - One of the best experts on this subject based on the ideXlab platform.
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Hybrid symplectic integrators for planetary dynamics
Monthly Notices of the Royal Astronomical Society, 2019Co-Authors: Hanno Rein, David M Hernandez, Garett Brown, Emily Eckels, Réjean Leblanc, Daniel Tamayo, Emma Holmes, Ari SilburtAbstract:Hybrid symplectic integrators such as MERCURY are widely used to simulate complex dynamical phenomena in planetary dynamics that could otherwise not be investigated. A hybrid integrator achieves high accuracy during close encounters by using a high order integration scheme for the duration of the encounter while otherwise using a standard 2nd order Wisdom-Holman scheme, thereby optimizing both speed and accuracy. In this paper we reassess the criteria for choosing the switching Function that determines which parts of the Hamiltonian are integrated with the high order integrator. We show that the original motivation for choosing a polynomial switching Function in MERCURY is not correct. We explain the nevertheless excellent performance of the MERCURY integrator and then explore a wide range of different switching Functions including an Infinitely Differentiable Function and a Heaviside Function. We find that using a Heaviside Function leads to a significantly simpler scheme compared to MERCURY, while maintaining the same accuracy in short term simulations.
E H Doha - One of the best experts on this subject based on the ideXlab platform.
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On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials
Journal of Physics A: Mathematical and General, 2004Co-Authors: E H DohaAbstract:Formulae expressing explicitly the Jacobi coefficients of a general-order derivative (integral) of an Infinitely Differentiable Function in terms of its original expansion coefficients, and formulae for the derivatives (integrals) of Jacobi polynomials in terms of Jacobi polynomials themselves are stated. A formula for the Jacobi coefficients of the moments of one single Jacobi polynomial of certain degree is proved. Another formula for the Jacobi coefficients of the moments of a general-order derivative of an Infinitely Differentiable Function in terms of its original expanded coefficients is also given. A simple approach in order to construct and solve recursively for the connection coefficients between Jacobi–Jacobi polynomials is described. Explicit formulae for these coefficients between ultraspherical and Jacobi polynomials are deduced, of which the Chebyshev polynomials of the first and second kinds and Legendre polynomials are important special cases. Two analytical formulae for the connection coefficients between Laguerre–Jacobi and Hermite–Jacobi are developed.
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On the connection coefficients and recurrence relations arising from expansions in series of hermite polynomials
Integral Transforms and Special Functions, 2004Co-Authors: E H DohaAbstract:A formula expressing the Hermite coefficients of a general-order derivative of an Infinitely Differentiable Function in terms of its original coefficients is proved, and a formula expressing explicitly the derivatives of Hermite polynomials of any degree and for any order as a linear combination of suitable Hermite polynomials is deduced. A formula for the Hermite coefficients of the moments of one single Hermite polynomial of certain degree is given. Formulae for the Hermite coefficients of the moments of a general-order derivative of an Infinitely Differentiable Function in terms of its Hermite coefficients are also obtained. Two numerical applications of how to use these formulae for solving ordinary differential equations with varying coefficients, by reducing them to recurrence relations in Hermite coefficients, are discussed. A simple approach in order to build and solve recursively for the connection coefficients between Jacobi–Hermite and Laguerre–Hermite polynomials is described. Explicit formula...
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On the connection coefficients and recurrence relations arising from expansions in series of Laguerre polynomials
Journal of Physics A: Mathematical and General, 2003Co-Authors: E H DohaAbstract:A formula expressing the Laguerre coefficients of a general-order derivative of an Infinitely Differentiable Function in terms of its original coefficients is proved, and a formula expressing explicitly the derivatives of Laguerre polynomials of any degree and for any order as a linear combination of suitable Laguerre polynomials is deduced. A formula for the Laguerre coefficients of the moments of one single Laguerre polynomial of certain degree is given. Formulae for the Laguerre coefficients of the moments of a general-order derivative of an Infinitely Differentiable Function in terms of its Laguerre coefficients are also obtained. A simple approach in order to build and solve recursively for the connection coefficients between Jacobi?Laguerre and Hermite?Laguerre polynomials is described. An explicit formula for these coefficients between Jacobi and Laguerre polynomials is given, of which the ultra-spherical polynomials of the first and second kinds and Legendre polynomials are important special cases. An analytical formula for the connection coefficients between Hermite and Laguerre polynomials is also obtained.
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On the coefficients of differentiated expansions and derivatives of Jacobi polynomials
Journal of Physics A: Mathematical and General, 2002Co-Authors: E H DohaAbstract:A formula expressing explicitly the derivatives of Jacobi polynomials of any degree and for any order in terms of the Jacobi polynomials themselves is proved. Another explicit formula, which expresses the Jacobi expansion coefficients of a general-order derivative of an Infinitely Differentiable Function in terms of its original Jacobi coefficients, is also given. The results for the special case of ultraspherical polynomials are considered. The results for Chebyshev polynomials of the first and second kinds and for Legendre polynomials are also noted. An application of how to use Jacobi polynomials for solving ordinary and partial differential equations is described.
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The ultraspherical coefficients of the moments of a general-order derivative of an Infinitely Differentiable Function
Journal of Computational and Applied Mathematics, 1998Co-Authors: E H DohaAbstract:Abstract A formula for the ultraspherical coefficients of the moments of one single ultraspherical polynomial of certain degree is given. Formulae for the ultraspherical coefficients of the moments of a general-order derivative of an Infinitely Differentiable Function in terms of its ultraspherical coefficients are also obtained. The corresponding formulae for the important special cases of Chebyshev polynomials of the first and second kinds and of Legendre polynomials are deduced. Two interesting numerical applications of how to use these formulae for solving ordinary differential equations with varying coefficients, by reducing them to recurrence relations of lowest order in the ultraspherical expansion coefficients, in the sense of Lewanowicz, are discussed. Comparisons with the results obtained by optimal algorithm of Lewanowicz (1976) are also made.
Hanno Rein - One of the best experts on this subject based on the ideXlab platform.
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Hybrid symplectic integrators for planetary dynamics
Monthly Notices of the Royal Astronomical Society, 2019Co-Authors: Hanno Rein, David M Hernandez, Garett Brown, Emily Eckels, Réjean Leblanc, Daniel Tamayo, Emma Holmes, Ari SilburtAbstract:Hybrid symplectic integrators such as MERCURY are widely used to simulate complex dynamical phenomena in planetary dynamics that could otherwise not be investigated. A hybrid integrator achieves high accuracy during close encounters by using a high order integration scheme for the duration of the encounter while otherwise using a standard 2nd order Wisdom-Holman scheme, thereby optimizing both speed and accuracy. In this paper we reassess the criteria for choosing the switching Function that determines which parts of the Hamiltonian are integrated with the high order integrator. We show that the original motivation for choosing a polynomial switching Function in MERCURY is not correct. We explain the nevertheless excellent performance of the MERCURY integrator and then explore a wide range of different switching Functions including an Infinitely Differentiable Function and a Heaviside Function. We find that using a Heaviside Function leads to a significantly simpler scheme compared to MERCURY, while maintaining the same accuracy in short term simulations.
David M Hernandez - One of the best experts on this subject based on the ideXlab platform.
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Hybrid symplectic integrators for planetary dynamics
Monthly Notices of the Royal Astronomical Society, 2019Co-Authors: Hanno Rein, David M Hernandez, Garett Brown, Emily Eckels, Réjean Leblanc, Daniel Tamayo, Emma Holmes, Ari SilburtAbstract:Hybrid symplectic integrators such as MERCURY are widely used to simulate complex dynamical phenomena in planetary dynamics that could otherwise not be investigated. A hybrid integrator achieves high accuracy during close encounters by using a high order integration scheme for the duration of the encounter while otherwise using a standard 2nd order Wisdom-Holman scheme, thereby optimizing both speed and accuracy. In this paper we reassess the criteria for choosing the switching Function that determines which parts of the Hamiltonian are integrated with the high order integrator. We show that the original motivation for choosing a polynomial switching Function in MERCURY is not correct. We explain the nevertheless excellent performance of the MERCURY integrator and then explore a wide range of different switching Functions including an Infinitely Differentiable Function and a Heaviside Function. We find that using a Heaviside Function leads to a significantly simpler scheme compared to MERCURY, while maintaining the same accuracy in short term simulations.
Garett Brown - One of the best experts on this subject based on the ideXlab platform.
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Hybrid symplectic integrators for planetary dynamics
Monthly Notices of the Royal Astronomical Society, 2019Co-Authors: Hanno Rein, David M Hernandez, Garett Brown, Emily Eckels, Réjean Leblanc, Daniel Tamayo, Emma Holmes, Ari SilburtAbstract:Hybrid symplectic integrators such as MERCURY are widely used to simulate complex dynamical phenomena in planetary dynamics that could otherwise not be investigated. A hybrid integrator achieves high accuracy during close encounters by using a high order integration scheme for the duration of the encounter while otherwise using a standard 2nd order Wisdom-Holman scheme, thereby optimizing both speed and accuracy. In this paper we reassess the criteria for choosing the switching Function that determines which parts of the Hamiltonian are integrated with the high order integrator. We show that the original motivation for choosing a polynomial switching Function in MERCURY is not correct. We explain the nevertheless excellent performance of the MERCURY integrator and then explore a wide range of different switching Functions including an Infinitely Differentiable Function and a Heaviside Function. We find that using a Heaviside Function leads to a significantly simpler scheme compared to MERCURY, while maintaining the same accuracy in short term simulations.