The Lie-group shooting method for singularly perturbed two-point boundary value problems

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 X 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 is an element of (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.