@Article{cmes.2006.015.179,
AUTHOR = {Chein-Shan Liu},
TITLE = {The Lie-Group Shooting Method for Singularly Perturbed Two-Point Boundary Value Problems},
JOURNAL = {Computer Modeling in Engineering \& Sciences},
VOLUME = {15},
YEAR = {2006},
NUMBER = {3},
PAGES = {179--196},
URL = {http://www.techscience.com/CMES/v15n3/26689},
ISSN = {1526-1506},
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 <i>x</i>-domain to the <i>t</i>-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 <i>r</i> ∈ (0,1) in the <i>t</i>-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 <i>r</i> 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.},
DOI = {10.3970/cmes.2006.015.179}
}