Modified Chebyshev-Picard Iteration Methods for Solution of Initial Value Problems

Bai, Xiaoli; Junkins, John L.

doi:10.1007/s40295-013-0021-6

Cited by 28 publications

(20 citation statements)

References 17 publications

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“…Two alternative methods for solving the perturbed Lambert's problem are the TPBVP implementation of modified Chebyshev-Picard iteration (MCPI) [3,4], and the MCPI-TPBVP algorithm regularized using the Kustaanheimo-Stiefel time transformation [5,6]. The first method converges over a small fraction of an orbit (about a third of an orbit depending on eccentricity) and the second method converges up to about 90% of an orbit.…”

Section: Introductionmentioning

confidence: 99%

Multiple Revolution Solutions for the Perturbed Lambert Problem using the Method of Particular Solutions and Picard Iteration

et al. 2017

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We present a new method for solving the multiple revolution perturbed Lambert problem using the method of particular solutions and modified ChebyshevPicard iteration. The method of particular solutions differs from the well-known Newton-shooting method in that integration of the state transition matrix (36 additional differential equations) is not required, and instead it makes use of a reference trajectory and a set of n particular solutions. Any numerical integrator can be used for solving two-point boundary problems with the method of particular solutions, however we show that using modified Chebyshev-Picard iteration affords an avenue for increased efficiency that is not available with other step-by-step integrators. We take advantage of the path approximation nature of modified Chebyshev-Picard iteration (nodes iteratively converge to fixed points in space) and utilize a variable fidelity force model for propagating the reference trajectory. Remarkably, we demonstrate that computing the particular solutions with only low fidelity function evaluations greatly increases the efficiency of the algorithm while maintaining machine precision accuracy. Our study reveals that solving the perturbed Lambert's problem using the method of particular solutions with modified Chebyshev-Picard iteration is about an order of magnitude faster compared with the classical shooting method and a tenthtwelfth order Runge-Kutta integrator. It is well known that the solution to Lambert's problem over multiple revolutions is not unique and to ensure that all possible solutions are considered we make use of a reliable preexisting Keplerian Lambert solver to warm start our perturbed algorithm.

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Section: Introductionmentioning

confidence: 99%

Multiple Revolution Solutions for the Perturbed Lambert Problem using the Method of Particular Solutions and Picard Iteration

et al. 2017

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“…To increase efficiency of STM calculations, a second-order differential equation may be used in place of Eq. 1; this method is called the MCPI Cascade Method [2]. The STM differential equation for the conservative case is rearranged to solve a pair of second-order equations as follows.…”

Section: Cascade Methodsmentioning

confidence: 99%

“…MCPI is an iterative, path approximation method for solving smoothly nonlinear systems of ordinary differential equations. The entire state trajectory over a long time arc is approximated at every iteration until a specified tolerance is met [2]. Emile Picard stated that, given an initial condition x (t 0 ) = x 0 , any first order differential equation [1] …”

Section: Modified Chebyshev Picard Iterationmentioning

confidence: 99%

“…For further details on the basics of this novel integration technique and its convergence properties, refer to References [1] and [2]. Since the Chebyshev polynomials are orthogonal, a matrix inverse is avoided when finding basis function coefficients.…”

Section: Modified Chebyshev Picard Iterationmentioning

confidence: 99%

“…A major advantage of the MCPI approach is that the use of cosine sampling reduces the well-known Runge Effect, whereby the largest errors are encountered near the boundary of piecewise approximation segments [1]. This algorithm has recently proven to be a powerful tool for solving nonlinear differential equations for both initial value problems (IVP) and boundary value problems (BVP) [1][2][3][4][5]. Comparisons with traditional step-by-step methods [6,7] have shown that MCPI is more efficient for a prescribed solution accuracy.…”

Section: Introductionmentioning

confidence: 99%

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State Transition Matrix for Perturbed Orbital Motion Using Modified Chebyshev Picard Iteration

et al. 2015

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The Modified Chebyshev Picard Iteration (MCPI) method has recently proven to be highly efficient for a given accuracy compared to several commonly adopted numerical integration methods, as a means to solve for perturbed orbital motion. This method utilizes Picard iteration, which generates a sequence of path approximations, and Chebyshev Polynomials, which are orthogonal and also enable both efficient and accurate function approximation. The nodes consistent with discrete Chebyshev orthogonality are generated using cosine sampling; this strategy also reduces the Runge effect and as a consequence of orthogonality, there is no matrix inversion required to find the basis function coefficients. The MCPI algorithms considered herein are parallel-structured so that they are immediately well-suited for massively parallel implementation with additional speedup. MCPI has a wide range of applications beyond ephemeris propagation, including the propagation of the State Transition Matrix (STM) for perturbed two-body motion. A solution is achieved for a spherical harmonic series representation of earth gravity (EGM2008), although the methodology is suitable for application to any gravity model. Included in this representation the normalized, Associated Legendre Functions are given and verified numerically. Modifications of the classical algorithm techniques, such as rewriting the STM equations in a second-order cascade formulation, gives rise to additional speedup. Timing results for the baseline formulation and this second-order formulation are given.

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Jacobi‐Picard iteration method for the numerical solution of nonlinear initial value problems

Tafakkori-Bafghi

Loghmani

Heydari

et al. 2019

Math Methods in App Sciences

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In this paper, an effective numerical iterative method for solving nonlinear initial value problems (IVPs) is presented. The proposed iterative scheme, called the Jacobi-Picard iteration (JPI) method, is based on the Picard iteration technique, orthogonal shifted Jacobi polynomials, and shifted Jacobi-Gauss quadrature formula. In comparison with traditional methods, the JPI method uses an iterative formula for updating next step approximations and calculating integrals of the shifted Jacobi polynomials are performed via an exact relation.Also, a vector-matrix form of the JPI method is provided in details which reduce the CPU time. The performance of the presented method has been investigated by solving several nonlinear IVPs. Numerical results show the efficiency and the accuracy of the proposed iterative method. KEYWORDS initial value problems, Jacobi-Gauss quadrature formula, Picard iteration method, shifted Jacobi polynomials, vector-matrix form MSC CLASSIFICATION 65L05; 33C45; 33F05; 65D32 | INTRODUCTIONOne of the scientific methods for solving physics, chemistry, and engineering problems and analyzing natural phenomena is constructing mathematical models for these problems. Some of these models arise in the form of initial value problems (IVPs), namely, a differential equation with the initial condition(s). Various methods are presented to solve IVPs. These methods are classified as analytical, semianalytical, and basically numerical methods.The most commonly used semianalytical methods are the Adomian decomposition method (ADM) and its modifications, 1-4 the variational iteration method (VIM), 5-8 the hybrid spectral-variational iteration method (H-S-VIM), 9 the homotopy perturbation method (HPM), 10-12 homotopy analysis method (HAM), 13,14 and the differential transformation method (DTM) 15-17 ; on the other hand, collocation method using Chebyshev and Legendre as Jacobi polynomials, 18-29 radial basis function method, 30-34 cubic B-spline scaling function and Chebyshev cardinal function method, [35][36][37] Birkhoff-type interpolation method, 38,39 collocation method using Bernstein polynomials, 40-43 the Galerkin method, 44 the hybrid block-pulse and Chebyshev cardinal function method, 45 the linear barycentric rational interpolation method, 46 and also other numerical methods such as Runge-Kutta-Nyström method and multistep predictor-corrector method are proposed by some researchers such as Ahmad et al, 47 Jator and Lee, 48 Kosti et al, 49,50 Papadopoulos et al, 51 and Simos 52 for solving periodical oscillatory IVPs.

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Modified Chebyshev-Picard Iteration Methods for Solution of Initial Value Problems

Cited by 28 publications

References 17 publications

Multiple Revolution Solutions for the Perturbed Lambert Problem using the Method of Particular Solutions and Picard Iteration

Multiple Revolution Solutions for the Perturbed Lambert Problem using the Method of Particular Solutions and Picard Iteration

State Transition Matrix for Perturbed Orbital Motion Using Modified Chebyshev Picard Iteration

Jacobi‐Picard iteration method for the numerical solution of nonlinear initial value problems

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