Second order global uniformly convergent numerical method for a coupled system of singularly perturbed initial value problems

We consider a coupled system of singularly perturbed first order ordinary differential equations. The equations have small positive parameters of different magnitudes associated with them. A decomposition of the exact solution into smooth and singular components is constructed that is useful for the analysis of the numerical method. We apply the HODIE (High Order Difference approximation with Identity Expansions) technique to construct a second order finite difference scheme and combine this with standard backward Euler difference scheme in a special way on a piecewise-uniform Shishkin mesh to solve the system numerically. It is proved that the method is second order (up to a logarithmic factor) convergent in the maximum norm uniformly in both perturbation parameters. Numerical experiments are conducted to support the theoretical results.