High‐order discrete‐time orthogonal spline collocation methods for singularly perturbed 1D parabolic reaction–diffusion problems

Abstract

AbstractQuasi‐optimal error estimates are derived for the continuous‐time orthogonal spline collocation (OSC) method and also two discrete‐time OSC methods for approximating the solution of 1D parabolic singularly perturbed reaction–diffusion problems. OSC with C1 splines of degree r ≥ 3 on a Shishkin mesh is employed for the spatial discretization while the Crank–Nicolson method and the BDF2 scheme are considered for the time‐stepping. The results of numerical experiments validate the theoretical analysis and also exhibit additional quasi‐optimal results, in particular, superconvergence phenomena.