An Adaptive Coordinate-Stretched Spectral Collocation–QLM Scheme for Singularly Perturbed Boundary-Layer Problems

Abstract

An adaptive coordinate-stretched spectral collocation algorithm with the quasilinearization technique is developed in this paper for the numerical solution of singularly perturbed boundary value problems. The method uses a stretching parameter in the computational domain, determined adaptively by a residual-based spectral criterion, thereby obviating the need for manual control over boundary layers. The approach is tested on a benchmark set of problems, including linear and nonlinear boundary-layer models. The scheme's accuracy is close to machine precision for the well-posed test cases. It remains stable and high-order accurate for strong singular perturbation regimes when sharp boundary layers are resolved using a modest degree of polynomials. Comparisons with the classical spectral collocation and fixed-mapped strategies are also presented, which show that the adaptive procedure both achieves node clustering results as good as those methods and yields competitive performance without the need for any specific ad hoc parameter tuning. Numerical results indicate the possibility of a simple and reliable alternative method for solving nonlinear boundary-layer problems.