Open Access

ARTICLE

The Lie-Group Shooting Method for Singularly Perturbed Two-Point Boundary Value Problems

Chein-Shan Liu¹

1 Department of Mechanical and Mechatronic Engineering, Taiwan Ocean University, Keelung, Taiwan. E-mail: csliu@mail.ntou.edu.tw

Computer Modeling in Engineering & Sciences 2006, 15(3), 179-196. https://doi.org/10.3970/cmes.2006.015.179

Abstract

This paper studies the numerical computations of the second-order singularly perturbed boundary value problems (SPBVPs). In order to depress the singularity we consider a coordinate transformation from the x-domain to the t-domain. The relation between singularity and stiffness is demonstrated, of which the coordinate transformation parameter λ plays a key role to balance these two tendencies. Then we construct a very effective Lie-group shooting method to search the missing initial condition through a weighting factor r ∈ (0,1) in the t-domain formulation. For stabilizing the new method we also introduce two new systems by a translation of the dependent variable. Numerical examples are examined to show that the new approach has high efficiency and high accuracy. Only through a few trials one can determine a suitable r very soon, and the new method can attain the second-order accuracy even for the highly singular cases. A finite difference method together with the nonstandard group preserving scheme for solving the resulting ill-posed equations is also provided, which is a suitable method for the calculations of SPBVPs without needing for many grid points. This method has the first-order accuracy.

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Keywords

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