A study of difference schemes with the first derivative approximated by a central difference ratio
Computational Mathematics and Mathematical Physics
Applied Numerical Mathematics
Pointwise convergence of approximations to a convection—diffusion equation on a Shishkin mesh
Applied Numerical Mathematics
A uniformly accurate spline collocation method for a normalized flux
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on boundary and interior layers - computational and asymptotic methods (BAIL 2002)
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We consider the finite difference approximation of a singularly perturbed one-dimensional convection-diffusion two-point boundary value problem. It is discretized using quadratic splines as approximation functions, equations with various piecewise constant coefficients as collocation equations and a piecewise uniform mesh of Shishkin type. The family of schemes is derived using the collocation method. The numerical methods developed here are non-monotone and therefore apart from the consistency error we use Green's grid function analysis to prove uniform convergence. We prove the almost first order of convergence and furthermore show that some of the schemes have almost second-order convergence. Numerical experiments presented in the paper confirm our theoretical results.