Second order parameter-uniform convergence for a finite difference method for a singularly perturbed linear reaction-diffusion system
For researchers solving singularly perturbed reaction-diffusion systems with multiple distinct parameters, this work provides a rigorously convergent numerical method.
The paper proves that a finite difference method on Shishkin meshes achieves essentially second-order uniform convergence for a singularly perturbed linear reaction-diffusion system with distinct small 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 singular perturbation 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 a numerical method for solving this problem. It is proved that the numerical approximations obtained with this method is essentially second order convergent uniformly with respect to all of the parameters.