A uniformly convergent difference scheme on a modified Shishkin mesh for the singular perturbation boundary value problem
Provides a robust numerical method for solving singularly perturbed boundary value problems with uniform accuracy, which is important for computational science but is an incremental improvement over existing Shishkin mesh techniques.
The paper constructs a difference scheme for semilinear singular perturbation reaction-diffusion problems and proves ε-uniform convergence of almost second order on a modified Shishkin mesh, supported by numerical examples.
In this paper we are considering a semilinear singular perturbation reaction -- diffusion boundary value problem, which contains a small perturbation parameter that acts on the highest order derivative. We construct a difference scheme on an arbitrary nonequidistant mesh using a collocation method and Green's function. We show that the constructed difference scheme has a unique solution and that the scheme is stable. The central result of the paper is $ε$-uniform convergence of almost second order for the discrete approximate solution on a modified Shishkin mesh. We finally provide two numerical examples which illustrate the theoretical results on the uniform accuracy of the discrete problem, as well as the robustness of the method.