A smooth locally-analytical technique for singularly perturbed two-point boundary-value problems

A locally-analytical method for singularly perturbed two-point boundary-value problems with internal and boundary layers and with turning points is presented. The method is based on the linearization of ordinary differential equations in nonoverlapping intervals and results in linear constant-coefficients ordinary differential equations which can be integrated analytically, thus yielding piecewise analytical solutions. By imposing continuity conditions at the end points of each interval plus a smoothness condition at the common end point of two adjacent intervals, a global smooth solution is obtained. The accuracy of the globally smooth locally-analytical method is assessed by comparisons with exact and approximate solutions of several singularly perturbed problems with internal and boundary layers. It is shown that the smooth locally-analytical method is more precise than second-order accurate finite difference discretizations. It is also shown that the accuracy of the smooth locally-analytical method depends on the kind of nonlinearity and inhomogeneities of singularly perturbed ordinary differential equations, but is always higher than that of exponentially-fitted techniques based on the local solution of advection-diffusion operators.