TY - EJOU
AU - Liu, Chein-Shan
TI - The Lie-Group Shooting Method for Singularly Perturbed Two-Point Boundary Value Problems
T2 - Computer Modeling in Engineering \& Sciences
PY - 2006
VL - 15
IS - 3
SN - 1526-1506
AB - 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.
KW - One-step group preserving scheme
KW - Singularly perturbed boundary value problem
KW - Boundary layer
KW - Lie-group shooting method
KW - Stiff equation
KW - Ill-posed equation
DO - 10.3970/cmes.2006.015.179