Linearization techniques for singularly-perturbed initial-value problems of ordinary differential equations

Linearization methods for singularly-perturbed initial-value problems in ordinary differential equations based on Taylor series expansions and piecewise analytical solutions, are presented. If only the linear terms are retained, it is shown that exponentially-fitted techniques are obtained, whereas, if quadratic terms are kept, then Riccati's equations result. Both linear and quadratic linearization methods are shown to be nonuniformly-convergent in the small perturbation parameter, and the former are more accurate than the latter in equally-spaced grids. Linearization methods are also used with three different adaptive mesh refinement methods that account for the magnitude of the Jacobian, the variation of the solution, and the resolution of the initial layer, and it is shown that mesh refinement methods based on the variation of the solution can provide very accurate solutions, although they may yield large errors at the outer edge of the initial layer if the grid spacing increases rapidly. It is also shown that, for values of the perturbation parameter on the order of unity, quadratic linearization methods are more accurate than linearization methods and can account for the presence of internal layers. It is also shown that linearization methods on grids whose spacing obeys a geometric progression are not robust and may suffer from a loss in the order of convergence compared with adaptive methods that resolve the boundary layer or adapt the grid according to the numerical solution.