TY - EJOU
AU - Liu, Chein-Shan
TI - Efficient Shooting Methods for the Second-Order Ordinary Differential Equations
T2 - Computer Modeling in Engineering \& Sciences
PY - 2006
VL - 15
IS - 2
SN - 1526-1506
AB - 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 **G(u**_{0},u_{T}) and the establishment of a mid-point Lie group element **G(***r*). Then, by imposing **G(u**_{0},u_{T}) = **G(***r*) we can search the missing initial conditions through an iterative solution of the weighting factor *r* ∈ (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.
KW - One-step group preserving scheme
KW - Boundary value problem
KW - Shooting method
KW - Estimation of missing initial condition
DO - 10.3970/cmes.2006.015.069