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

doi:10.3970/cmes.2006.015.179
Source CMES: Computer Modeling in Engineering & Sciences, Vol. 15, No. 3, pp. 179-196, 2006
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Keywords One-step group preserving scheme, Singularly perturbed boundary value problem, Boundary layer, Lie-group shooting method, Stiff equation, Ill-posed equation.
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$\lambda$ 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$r \in (0,1)$ 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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