A stepsize control for continuation methods and its special application to multiple shooting techniques

1979 ◽  
Vol 33 (2) ◽  
pp. 115-146 ◽  
Author(s):  
P. Deuflhard
Keyword(s):  
Continuation Methods ◽  
Multiple Shooting ◽  
Stepsize Control ◽  
Multiple Shooting Techniques

scholarly journals Three-Step Block Method for Solving Nonlinear Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Phang Pei See ◽  
Zanariah Abdul Majid ◽  
Mohamed Suleiman
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Second Order ◽  
Multiple Shooting ◽  
Variable Step Size ◽  
Nonlinear Boundary Value Problems ◽  
Block Method ◽  
Step Size ◽  
Multiple Shooting Techniques

We propose a three-step block method of Adam’s type to solve nonlinear second-order two-point boundary value problems of Dirichlet type and Neumann type directly. We also extend this method to solve the system of second-order boundary value problems which have the same or different two boundary conditions. The method will be implemented in predictor corrector mode and obtain the approximate solutions at three points simultaneously using variable step size strategy. The proposed block method will be adapted with multiple shooting techniques via the three-step iterative method. The boundary value problem will be solved without reducing to first-order equations. The numerical results are presented to demonstrate the effectiveness of the proposed method.

scholarly journals Low-thrust Lyapunov to Lyapunov and Halo to Halo missions with L2-minimization

2017 ◽  
Vol 51 (3) ◽  
pp. 965-996 ◽  
Author(s):  
Maxime Chupin ◽  
Thomas Haberkorn ◽  
Emmanuel Trélat
Keyword(s):  
Periodic Orbits ◽  
Invariant Manifolds ◽  
Heteroclinic Orbit ◽  
Continuation Methods ◽  
Heteroclinic Orbits ◽  
Multiple Shooting ◽  
Shooting Methods ◽  
Low Thrust ◽  
Lyapunov Orbits ◽  
Multiple Shooting Method

In this work, we develop a new method to design energy minimum low-thrust missions (L2-minimization). In the Circular Restricted Three Body Problem, the knowledge of invariant manifolds helps us initialize an indirect method solving a transfer mission between periodic Lyapunov orbits. Indeed, using the PMP, the optimal control problem is solved using Newton-like algorithms finding the zero of a shooting function. To compute a Lyapunov to Lyapunov mission, we first compute an admissible trajectory using a heteroclinic orbit between the two periodic orbits. It is then used to initialize a multiple shooting method in order to release the constraint. We finally optimize the terminal points on the periodic orbits. Moreover, we use continuation methods on position and on thrust, in order to gain robustness. A more general Halo to Halo mission, with different energies, is computed in the last section without heteroclinic orbits but using invariant manifolds to initialize shooting methods with a similar approach.

scholarly journals New Algorithm of Two-Point Block Method for Solving Boundary Value Problem with Dirichlet and Neumann Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Pei See Phang ◽  
Zanariah Abdul Majid ◽  
Fudziah Ismail ◽  
Khairil Iskandar Othman ◽  
Mohamed Suleiman
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Multiple Shooting ◽  
Variable Step Size ◽  
Block Method ◽  
Step Size ◽  
Multiple Shooting Techniques ◽  
Neumann Type

Two-point block method with variable step-size strategy is presented to obtain the solutions for boundary value problems directly. Dirichlet type and Neumann type of boundary conditions are studied in this paper. Multiple shooting techniques adapted with the three-step iterative method are employed for generating the guessing value. Six boundary value problems are solved using the proposed method, and the numerical results are compared to the existing methods. The results suggest a significant improvement in the efficiency of the proposed methods in terms of the number of steps, execution time, and accuracy.

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