A parameter--uniform finite difference method for a singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type
Provides provably convergent numerical methods for a class of parameter-dependent boundary value problems, which is incremental as it extends existing Shishkin mesh techniques to a specific system.
The paper develops finite difference methods for a singularly perturbed system of reaction-diffusion ODEs with distinct small parameters, achieving first- and second-order uniform convergence in all parameters.
A singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with given boundary conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are assumed to be distinct. The components of the solution exhibit overlapping layers. Shishkin piecewise-uniform meshes are introduced, which are used in conjunction with a classical finite difference discretisation, to construct two numerical methods for solving this problem. It is proved that the numerical approximations obtained with these methods are essentially first, respectively second, order convergent uniformly with respect to all of the parameters.