Collocation Schemes for Quasilinear Singularly Perturbed Boundary Value Problems.

摘要

Many high-order discretization methods for the solution of two-point boundary value problems for systems of ordinary differential equations are already circulated. However, these methods can behave quite poorly in case the solution has large derivatives, unless severe restrictions on the mesh are imposed. The reason for these restrictions is mainly a stability problem. In this paper a strongly A-stable difference method based on polynomial collocation is developed for a class of qusilinear, singularly perturbed, two-point boundary value problems. Many problems of practical interest asre included in this class, for instance the nonlinear deformation of thin beams or one-dimensional models of carrier transport in semiconductor devices. The method combines the advantages of having the same stability properties as the lower order methods which are used already, with the high order of convergence of collocation methods. It is shown that the number of gridpoints (and therefore the amount of computing time and required storage) is of the same order of magnitude as the one required for solving unperturbed problems.