Efficient Shooting Methods for the Second-Order Ordinary Differential Equations
Chein-Shan Liu

doi:10.3970/cmes.2006.015.069
Source CMES: Computer Modeling in Engineering & Sciences, Vol. 15, No. 2, pp. 69-86, 2006
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Keywords One-step group preserving scheme, Boundary value problem, Shooting method, Estimation of missing initial condition.
Abstract In this paper we will study the numerical integrations of second order boundary value problems under the imposed conditions at$t=0$ and$t=T$ in a general setting. We can construct a compact space shooting method for finding the unknown initial conditions. The key point is based on the construction of a one-step Lie group element${\bf G}({\bf u}_0,{\bf u}_T)$ and the establishment of a mid-point Lie group element${\bf G}(r)$. Then, by imposing${\bf G}({\bf u}_0,{\bf u}_T)={\bf G}(r)$ we can search the missing initial conditions through an iterative solution of the weighting factor$r \in (0,1)$. Numerical examples were examined to convince that the new approach has high efficiency and accuracy with a fast convergence speed by solving$r$ with a half-interval method. Even under a large span of the boundary coordinate, the new method is also applicable by requiring only a few iterations. The method is also extended to the BVP with general boundary conditions.
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