Singular perturbation problems in ordinary differential equations can pose significant numerical difficulties. Those problems often arise in many practical applications such as semiconductor theory, fluid dynamics, seismology and nonlinear mechanics. Often in such problems one may expect some high-order derivatives to be multiplied by a small parameter \(\epsilon\). In discretizing a singularly perturbed problem, the large local variations of the derivatives usually suggests a highly nonuniform mesh that is fine when the solution varies rapidly and coarse where the solution is smooth.
If one calls h the largest mesh stepsize used over the entire domain, then for reasons of efficiency \(\epsilon \ll h\) that means h cannot be chosen small with respect to the problem coefficient \(\epsilon\). This situation raises theoretical difficulties because the usual asymptotic theory, which is valid when h can be chosen “sufficiently small” no longer holds. Hence one has two major difficulties: (a) the choice of a suitable discretization method and a theory to support its use, (b) the construction of an appropriate nonuniform mesh.
The authors discuss boundary value problems of the singularly perturbed type for linear first-order systems. One basic difficulty in designing suitable discretization methods for this class of problems is the fact that the differential operators considered give raise to various modes with very different behaviour types: one may have rapidly increasing and rapidly decreasing fundamental solutions as well as harmless slow ones. One general approach to solve problems with mixed solution types is to attempt to find numerical schemes that are capable at least to some acceptable degree of simultaneously handling the various solution types. For the above boundary value problems piecewise polynomial collocation at Gaussian points provides a family of schemes belonging to the mentioned approach.
In several earlier papers the first author has exclusively analyzed, implemented and tested this method. In the present note the authors propose an improved algorithm for adaptive mesh selection using collocation at Gaussian points. They obtain efficient high-order approximations to extremely stiff problems. These modifications work much better than earlier implementations as the singular perturbation parameter gets small i.e. the problem gets stiff.