Block variable order step size method for solving higher order orbital problems

The current numerical techniques for solving a system of higher order ordinary differential equations (ODEs) directly calculate the integration coefficients at every step. Here, we propose a method to solve higher order ODEs directly by calculating the integration coefficients only once at the beginning of the integration and if required once more at the end. The formulae will be derived in terms of backward difference in a constant step size formulation. The method developed will be validated by solving some higher order ODEs directly using variable order step size. To simplify the evaluations of the integration coefficients, we find the relationship between various orders. The results presented confirmed our hypothesis.

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Variable-step-size second-order-derivative multistep method for solving first-order ordinary differential equations in system simulation

International Journal of Modeling Simulation and Scientific Computing ◽

10.1142/s1793962320500014 ◽

2020 ◽

Vol 11 (01) ◽

pp. 2050001

Author(s):

Lei Zhang ◽

Chaofeng Zhang ◽

Mengya Liu

Keyword(s):

Differential Equations ◽

Numerical Method ◽

Multistep Method ◽

Second Order ◽

Order Derivative ◽

Variable Step Size ◽

Variable Order ◽

Step Size ◽

Variable Step ◽

The Stability

According to the relationship between truncation error and step size of two implicit second-order-derivative multistep formulas based on Hermite interpolation polynomial, a variable-order and variable-step-size numerical method for solving differential equations is designed. The stability properties of the formulas are discussed and the stability regions are analyzed. The deduced methods are applied to a simulation problem. The results show that the numerical method can satisfy calculation accuracy, reduce the number of calculation steps and accelerate calculation speed.

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CpG Island Identification with Higher Order and Variable Order Markov Models

Data Mining in Biomedicine - Springer Optimization and Its Applications ◽

10.1007/978-0-387-69319-4_4 ◽

2008 ◽

pp. 47-57

Author(s):

Zhenqiu Liu ◽

Dechang Chen ◽

Xue-wen Chen

Keyword(s):

Markov Models ◽

Cpg Island ◽

Higher Order ◽

Variable Order

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Variable order and variable step-size integration method for transient analysis programs

IEEE Transactions on Power Systems ◽

10.1109/59.131064 ◽

1991 ◽

Vol 6 (1) ◽

pp. 206-213 ◽

Cited By ~ 9

Author(s):

T. Kato ◽

K. Ikeuchi

Keyword(s):

Transient Analysis ◽

Integration Method ◽

Variable Step Size ◽

Variable Order ◽

Step Size ◽

Variable Step

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Free Vibration Response of Thin and Thick Nonhomogeneous Shells by Refined One-Dimensional Analysis

Journal of Vibration and Acoustics ◽

10.1115/1.4028127 ◽

2014 ◽

Vol 136 (6) ◽

Cited By ~ 7

Author(s):

Alberto Varello ◽

Erasmo Carrera

Keyword(s):

Finite Element ◽

Free Vibration ◽

Cross Section ◽

Vibration Analysis ◽

Plane Deformation ◽

Free Vibration Analysis ◽

Higher Order ◽

Variable Order ◽

Vibrational Modes ◽

One Dimensional

The free vibration analysis of thin- and thick-walled layered structures via a refined one-dimensional (1D) approach is addressed in this paper. Carrera unified formulation (CUF) is employed to introduce higher-order 1D models with a variable order of expansion for the displacement unknowns over the cross section. Classical Euler–Bernoulli (EBBM) and Timoshenko (TBM) beam theories are obtained as particular cases. Different kinds of vibrational modes with increasing half-wave numbers are investigated for short and relatively short cylindrical shells with different cross section geometries and laminations. Numerical results of natural frequencies and modal shapes are provided by using the finite element method (FEM), which permits various boundary conditions to be handled with ease. The analyses highlight that the refinement of the displacement field by means of higher-order terms is fundamental especially to capture vibrational modes that require warping and in-plane deformation to be detected. Classical beam models are not able to predict the realistic dynamic behavior of shells. Comparisons with three-dimensional elasticity solutions and solid finite element solutions prove that CUF provides accuracy in the free vibration analysis of even short, nonhomogeneous thin- and thick-walled shell structures, despite its 1D approach. The results clearly show that bending, radial, axial, and also shell lobe-type modes can be accurately evaluated by variable kinematic 1D CUF models with a remarkably lower computational effort compared to solid FE models.

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Massively parallel structured direct solver for equations describing time-harmonic qP-polarized waves in TTI media

Geophysics ◽

10.1190/geo2011-0163.1 ◽

2012 ◽

Vol 77 (3) ◽

pp. T69-T82 ◽

Cited By ~ 15

Author(s):

Shen Wang ◽

Jianlin Xia ◽

Maarten V. de Hoop ◽

Xiaoye S. Li

Keyword(s):

Finite Difference ◽

Higher Order ◽

Difference Schemes ◽

Massively Parallel ◽

Finite Difference Schemes ◽

Variable Order ◽

Nested Dissection ◽

Partial Differential ◽

Polarized Waves ◽

Time Harmonic

We considered the discretization and approximate solutions of equations describing time-harmonic qP-polarized waves in 3D inhomogeneous anisotropic media. The anisotropy comprises general (tilted) transversely isotropic symmetries. We are concerned with solving these equations for a large number of different sources. We considered higher-order partial differential equations and variable-order finite-difference schemes to accommodate anisotropy on the one hand and allow higher-order accuracy — to control sampling rates for relatively high frequencies — on the other hand. We made use of a nested dissection based domain decomposition in a massively parallel multifrontal solver combined with hierarchically semiseparable matrix compression techniques. The higher-order partial differential operators and the variable-order finite-difference schemes require the introduction of separators with variable thickness in the nested dissection; the development of these and their integration with the multifrontal solver is the main focus of our study. The algorithm that we developed is a powerful tool for anisotropic full-waveform inversion.

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Higher-Order Weak Approximation of Ito Diffusions by Markov Chains

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800002606 ◽

1992 ◽

Vol 6 (3) ◽

pp. 391-408 ◽

Cited By ~ 10

Author(s):

Eckhaard Platen

Keyword(s):

Markov Chains ◽

Diffusion Process ◽

Transition Probabilities ◽

Higher Order ◽

Weak Order ◽

Weak Approximation ◽

Step Size ◽

Discrete State

This paper proposes a method that allows the construction of discrete-state Markov chains approximating an Ito-diffusion process. The transition probabilities of the Markov chains are chosen in such a way that functionals converge with a desired weak order with respect to vanishing step size under sufficient smoothness assumptions.

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Development of the Finite Segment Method for Modeling Railroad Track Structures

2011 Joint Rail Conference ◽

10.1115/jrc2011-56099 ◽

2011 ◽

Author(s):

Martin B. Hamper ◽

Khaled E. Zaazaa ◽

Ahmed A. Shabana

Keyword(s):

Sparse Matrix ◽

Deformable Body ◽

Contact Point ◽

Equations Of Motion ◽

Variable Step Size ◽

Variable Order ◽

Step Size ◽

Finite Segment ◽

Finite Segment Method ◽

Railroad Vehicle

In the finite segment method, the dynamics of a deformable body is described using a set of rigid bodies that are connected by elastic force elements. This approach can be used, as demonstrated in this investigation, in the simulation of some rail movement scenarios. The purpose of this investigation is to develop a new track model that combines the absolute nodal coordinate formulation (ANCF) geometry and the finite segment method. The ANCF finite elements define the track geometry in the reference configuration as well as the change in the geometry due to the movement of the finite segments of the track. Using ANCF geometry and the finite segment kinematics, the location of the wheel/rail contact point is predicted online and used to update the creepage expressions due to the finite segment displacements and rotations. The location of the wheel/rail contact point and the updated creepage expressions are used to evaluate the creep forces. A three-dimensional elastic contact formulation (ECF-A), that allows for wheel/rail separation, is used in this investigation. The rail displacement due to the applied loads is modeled by a set of rigid finite segments that are connected by set of spring-damper elements. Each rail finite segment is assumed to have six rigid body degrees of freedom. The equations of motion of the finite segments are integrated with the railroad vehicle system equations of motion in a sparse matrix formulation. The resulting dynamic equations are solved using a predictor-corrector numerical integration method that has a variable order and variable step size. As shown in this paper, the finite segments may be used to model specific phenomena that occur in railroad vehicle applications, including rail rotations and gage widening. The procedure used in this investigation to implement the finite segment method in a general purpose multibody system (MBS) computer program is described. Four simple models are presented in order to demonstrate the implementation of the finite segment method in MBS algorithms. The limitations of using the finite segments approach for modeling the track structure and rail flexibility are also discussed.

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