Numerical investigation for solutions and derivatives of singularly perturbed initial value problems

Abstract

This article proposes a hybrid scheme on layer-adapted meshes for solving singularly perturbed initial value problem depending on a parameter. Layer-adapted meshes namely standard Shishkin mesh and modified Shishkin mesh (Bakhvalov-Shishkin mesh and Vulanovic mesh) are considered. The hybrid scheme is a combination of second order central difference scheme on the fine mesh and a modified midpoint upwind scheme on the coarse mesh. The error analysis is carried out. We establish a second order parameter uniform convergence rate for the numerical solution and also for the scaled numerical derivative. It is also shown that the modified Shishkin mesh and graded mesh like Gartland-Shishkin mesh and Duran-Shishkin mesh give better results than the standard Shishkin mesh. In order to illustrate the efficiency of the proposed method, some numerical experiments are shown which support the theoretical findings.