An almost third order finite difference scheme for singularly perturbed reaction–diffusion systems

Abstract

a b s t r a c tThis paper addresses the numerical approximation of solutions to coupled systems of singularly perturbed reaction-diffusion problems. In particular a hybrid finite difference scheme of HODIE type is constructed on a piecewise uniform Shishkin mesh. It is proved that the numerical scheme satisfies a discrete maximum principle and also that it is third order (except for a logarithmic factor) uniformly convergent, even for the case in which the diffusion parameter associated with each equation of the system has a different order of magnitude. Numerical examples supporting the theory are given.

Optimal error estimate using mesh equidistribution technique for singularly perturbed system of reaction–diffusion boundary-value problems

“…However, it is well-known that one can have at most OðN 2 ln 2 ðNÞÞ convergence for reaction-diffusion problems on Shishkin mesh. It should also be noted that the higher-order numerical methods for system of reaction-diffusion problems are carried out by Clavero et al [5] and Melenk et al [19] and for system of Robin type reaction-diffusion problems by Das and Natesan [7] using hybrid scheme on piecewise-uniform Shishkin mesh.…”

An efficient parameter uniform spline-based technique for singularly perturbed weakly coupled reaction-diffusion systems

“…They have shown that the scheme is parameter-uniformly convergent of orders two and three in the cases of different and equal diffusion parameters, respectively. They have also addressed a hybrid FDM of HODIE type on a piecewise uniform Shishkin mesh for the coupled systems of singularly perturbed reaction-diffusion equations [6]. They have shown that the discretized operator satisfies the discrete maximum principle, and the scheme is almost third-order parameter-uniformly convergent (except for a logarithmic factor).…”

An efficient hybrid numerical method based on an additive scheme for solving coupled systems of singularly perturbed linear parabolic problems

“…For a stationary version of (1), a high order numerical scheme on a Shishkin mesh is developed in. 30 Note that the analysis of that scheme is not easy, it requires special barrier functions, and the accuracy of the method is proven to be O(N −3 ln 4 N). Nevertheless, for a stationary version of (1), the present scheme on generalized Shishkin mesh can prove to be O(N −4 ln N + N −4 ln 4 N + N −5 ln 3 N), which is better than the earlier mentioned convergence rate.…”