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# Shooting method

About: Shooting method is a research topic. Over the lifetime, 5048 publications have been published within this topic receiving 72238 citations.

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TL;DR: In this paper, the boundary value problem is discretized on several grids (or finite-element spaces) of widely different mesh sizes, and interactions between these levels enable us to solve the possibly nonlinear system of n discrete equations in 0(n) operations (40n additions and shifts for Poisson problems); and conveniently adapt the discretization (the local mesh size, local order of approximation, etc.) to the evolving solution in a nearly optimal way, obtaining "°°-order" approximations and low n, even when singularities are present.

Abstract: The boundary-value problem is discretized on several grids (or finite-element spaces) of widely different mesh sizes. Interactions between these levels enable us (i) to solve the possibly nonlinear system of n discrete equations in 0(n) operations (40n additions and shifts for Poisson problems); (ii) to conveniently adapt the discretization (the local mesh size, local order of approximation, etc.) to the evolving solution in a nearly optimal way, obtaining \"°°-order\" approximations and low n, even when singularities are present. General theoretical analysis of the numerical process. Numerical experiments with linear and nonlinear, elliptic and mixed-type (transonic flow) problemsconfirm theoretical predictions. Similar techniques for initial-value problems are briefly

2,923 citations

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TL;DR: A condensing algorithm for the solution of the approximating linearly constrained quadratic subproblems, and high rank update procedures are introduced, which are especially suited for optimal control problems and lead to significant improvements of the convergence behaviour and reductions of computing time and storage requirements.

Abstract: An algorithm for the numerical solution of parameterized optimal control problems is presented, which is based on multiple shooting in connection with a recursive quadratic progrmrming technique. A condensing algorithm for the solution of the approximating linearly constrained quadratic subproblems, and high rank update procedures are introduced, which are especially suited for optimal control problems and lead to significant improvements of the convergence behaviour and reductions of computing time and storage requirements. The algorithm is completely derivative-free due to internal numerical differentiation schemes, it can be conveniently combined with indirect multiple shooting. Numerical results are given in the field of aerospace engineering.

1,148 citations

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01 Jan 1985

TL;DR: This book discusses Consistency, Stability, and Convergence higher-Order One-Step Schemes Collocation Theory Acceleration Techniques Higher-Order ODEs Finite Element Methods and Initial Value Methods.

Abstract: List of Examples Preface 1. Introduction. Boundary Value Problems for Ordinary Differential Equations Boundary Value Problems in Applications 2. Review of Numerical Analysis and Mathematical Background. Errors in Computation Numerical Linear Algebra Nonlinear Equations Polynomial Interpolation Piecewise Polynomials, or Splines Numerical Quadrature Initial Value Ordinary Differential Equations Differential Operators and Their Discretizations 3. Theory of Ordinary Differential Equations. Existence and Uniqueness Results Green's Functions Stability of Initial Value Problems Conditioning of Boundary Value Problems 4. Initial Value Methods. Introduction. Shooting Superposition and Reduced Superposition Multiple Shooting for Linear Problems Marching Techniques for Multiple Shooting The Riccati Method Nonlinear Problems 5. Finite Difference Methods. Introduction Consistency, Stability, and Convergence Higher-Order One-Step Schemes Collocation Theory Acceleration Techniques Higher-Order ODEs Finite Element Methods 6. Decoupling. Decomposition of Vectors Decoupling of the ODE Decoupling of One-Step Recursions Practical Aspects of Consistency Closure and Its Implications 7. Solving Linear Equations. General Staircase Matrices and Condensation Algorithms for the Separated BC Case Stability for Block Methods Decomposition in the Nonseparated BC Case Solution in More General Cases 8. Solving Nonlinear Equations. Improving the Local Convergence of Newton's Method Reducing the Cost of the Newton Iteration Finding a Good Initial Guess Further Remarks on Discrete Nonlinear BVPS 9. Mesh Selection. Introduction Direct Methods A Mesh Strategy for Collocation Transformation Methods General Considerations 10. Singular Perturbations. Analytical Approaches Numerical Approaches Difference Methods Initial Value Methods 11. Special Topics. Reformulation of Problems in 'Standard' Form Generalized ODEs and Differential Algebraic Equations Eigenvalue Problems BVPs with Singularities Infinite Intervals Path Following, Singular Points and Bifurcation Highly Oscillatory Solutions Functional Differential Equations Method of Lines for PDEs Multipoint Problems On Code Design and Comparison Appendix A. A Multiple Shooting Code Appendix B. A Collocation Code References Bibliography Index.

1,115 citations

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TL;DR: Wazwaz et al. as mentioned in this paper applied homotopy perturbation method to nonlinear boundary value problems and compared the result obtained by the present method with that obtained by Adomian method.

Abstract: Homotopy perturbation method is applied to nonlinear boundary value problems. Comparison of the result obtained by the present method with that obtained by Adomian method [A.M. Wazwaz, Found. Phys. Lett. 13 (2000) 493] reveals that the present method is very effective and convenient.

1,054 citations

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756 citations