The Richardson extrapolation technique for quasilinear parabolic singularly perturbed convection-diffusion equations

Abstract

Abstract. The Dirichlet problem is considered for a quasilinear singularly perturbed parabolic convection-diffusion equation on a rectangular domain. For this problem classical finite difference (nonlinear) schemes on piecewise uniform meshes condensing in the boundary layer converge ε-uniformly at a rate that is at best first-order. Using a Richardson extrapolation technique, we construct an improved (nonlinear) scheme that is ε-uniformly convergent at the, where N and N0 define the number of nodes in the spatial and time meshes, respectively. This nonlinear scheme is used in the construction of a linearized scheme where at each time level the nonlinear term is evaluated using the computed solution from the previous time level. Furthermore, using the linearized and nonlinear improved schemes, we construct a linearized improved Richardson scheme that converges ε-uniformly at the rate O((N −1 ln N ) 2 + N −q 0 ), where q ≥ 2.

Performance of Richardson Extrapolation on Some Numerical Methods for a Singularly Perturbed Turning Point Problem Whose Solution Has Boundary Layers

“…However, the rates of convergence of these methods remain unimproved. This is in disagreement with a statement found in [27]. These conclusions drawn from our analysis are further confirmed by the numerical results found in tables 1-12.…”

“…In order to achieve better accuracy, we compute two numerical solutions via our FOFDM on two but nested meshes. We show that with a suitable choice of a linear combination of these two solutions, one can improve the accuracy and the order of convergence [6,27].…”

“…A linear combination of these approximations yields a better accuracy as compared to when each of them taken separately. It is also expected that this procedure will improve the rates of convergence of these numerical approximations [27]. In the past, an almost first order convergence is improved to an almost second order convergence in [21] via extrapolation based on fitted mesh finite difference method for a convection diffusion problem.…”

“…Note that, in all the cases, Richardson extrapolation improves the accuracy of the underlying method. For more details about Richardson extrapolation, the interested reader are referred to [6,12,13,24,26,27].…”

Limitations of Richardson’s extrapolation for a high order fitted mesh method for self-adjoint singularly perturbed problems

“…Several methods for improvement of solutions have been designed (see for example Andreev and Popov [1], Gaunt [4], Joyce [7], Richardson [15], Shishkin [16], Shishkin and Shishkina [17] and the references therein). One of these methods, which was subsequently termed as Richardson extrapolation, is a post-processing procedure where a linear combination of two computed solutions approximating a particular quantity gives a third and better approximation.…”