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High‐order discrete‐time orthogonal spline collocation methods for singularly perturbed 1D parabolic reaction–diffusion problems
Author(s) -
Mishra Pankaj,
Sharma Kapil K.,
Pani Amiya K.,
Fairweather Graeme
Publication year - 2020
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22438
Subject(s) - superconvergence , mathematics , crank–nicolson method , discretization , reaction–diffusion system , spline (mechanical) , orthogonal collocation , collocation (remote sensing) , collocation method , mathematical analysis , finite element method , differential equation , ordinary differential equation , physics , structural engineering , engineering , thermodynamics , remote sensing , geology
Quasi‐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 C 1 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.