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

Read, Julie L.; Younes, Ahmad Bani; Macomber, Brent; Turner, James D.; Junkins, John L.

doi:10.1007/s40295-015-0051-3

Cited by 19 publications

(12 citation statements)

References 10 publications

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“…(1) can be solved by many existing approaches, most of which are based on the Runge-Kutta family of integrators [1]. Other methods include: Gauss-Jackson [2], time domain collocation techniques [3,4], and Modified Chebyshev-Picard Iteration [5][6][7] (a path-length integral approximation which has been recently proven to be highly effective). All of these methods are based on low-order Taylor expansions, which limit the step size that can be used to propagate the solution.…”

Section: Introductionmentioning

confidence: 99%

High accuracy least-squares solutions of nonlinear differential equations

Mortari

Johnston

Smith

2019

Journal of Computational and Applied Mathematics

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This study shows how to obtain least-squares solutions to initial and boundary value problems of ordinary nonlinear differential equations. The proposed method begins using an approximate solution obtained by any existing integrator. Then, a least-squares fitting of this approximate solution is obtained using a constrained expression, derived from Theory of Connections. In this expression, the differential equation constraints are embedded and are always satisfied. The resulting constrained expression is then used as an initial guess in a Newton iterative process that increases the solution accuracy to machine error level in no more than two iterations for most of the problems considered. An analysis of speed and accuracy has been conducted for this method using two nonlinear differential equations. For non-smooth solutions or for long integration times, a piecewise approach is proposed. The highly accurate value estimated at the final time is then used as the new initial guess for the next time range, and this process is repeated for subsequent time ranges. This approach has been applied and validated solving the Duffing oscillator obtaining a final solution error on the order of 10 −12. To complete the study, a final numerical test is provided for a boundary value problem with a known solution.

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Section: Introductionmentioning

confidence: 99%

High accuracy least-squares solutions of nonlinear differential equations

Mortari

Johnston

Smith

2019

Journal of Computational and Applied Mathematics

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“…The absolute error in the symplectic nature of the J 2 −J 6 perturbed STM is calculated by Eq. 45, [25] […”

Section: Symplectic Error For the Gravity Perturbed Stmmentioning

confidence: 99%

“…In this method, first and second order STM is decoupled from the state propagation to make the procedure computationally efficient. To formulate the STM with Spherical Harmonic Gravity model, Read et al applied Modified Chebyshev Picard Iteration method, [25]. The method is shown to be well suited for parallel implementation for additional speed of computation.…”

Section: Introductionmentioning

confidence: 99%

An Adaptive Analytic Continuation Method for Computing the Perturbed Two-Body Problem State Transition Matrix

Tasif

Elgohary

2020

J Astronaut Sci

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In this work, the Taylor series based technique, Analytic Continuation is implemented to develop a method for the computation of the gravity and drag perturbed State Transition Matrix (STM) incorporating adaptive time steps and expansion order. Analytic Continuation has been developed for the two-body problem based on two scalar variables f and gp and their higher order time derivatives using Leibniz rule. The method has been proven to be very precise and efficient in trajectory propagation. The method is expanded to include the computation of the STM for the perturbed two-body problem. Leibniz product rule is used to compute the partials for the recursive formulas and an arbitrary order Taylor series is used to compute the STM. Four types of orbits, LEO, MEO, GTO and HEO, are presented and the simulations are run for 10 orbit periods. The accuracy of the STM is evaluated via RMS error for the unperturbed cases, symplectic check for the gravity perturbed cases and error propagation for the gravity and drag perturbed orbits. The results are compared against analytical and high order numerical solvers (ODE45, ODE113 and ODE87) in terms of accuracy. The results show that the method maintains double-precision accuracy for all test cases and 1-2 orders of magnitude improvement in linear prediction results compared to ODE87. The present approach is simple, adaptive and can readily be expanded to compute the full spherical harmonics gravity perturbations as well as the higher order state transition tensors.

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“…Also, a slight modification of CPM was introduced by Junkins et al . Thereafter, some researchers used the MCPI method for solving astronautical problems …”

Section: Introductionmentioning

confidence: 99%

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

Cited by 19 publications

References 10 publications

High accuracy least-squares solutions of nonlinear differential equations

High accuracy least-squares solutions of nonlinear differential equations

An Adaptive Analytic Continuation Method for Computing the Perturbed Two-Body Problem State Transition Matrix

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

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