Second order parameter-uniform numerical method for a partially singularly perturbed linear system of reaction-diusion type

A partially singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with given boundary conditions is considered. The leading terms of first $m$ equations are multiplied by small positive singular perturbation parameters which are assumed to be distinct. The rest of the equations are not singularly perturbed. The first $m$ components of the solution exhibit overlapping layers and the remaining $n-m$ components have less-severe overlapping layers. Shishkin piecewise-uniform meshes are used in conjunction with a classical finite difference discretisation, to construct a numerical method for solving this problem. It is proved that the numerical approximation obtained by this method is essentially second order convergent uniformly with respect to all the parameters. Numerical illustrations are presented in support of the theory.