Second order parameter-uniform convergence for a finite difference method for a singularly perturbed linear parabolic system
This work provides a rigorous convergence analysis for a numerical method handling overlapping layers in a system of reaction-diffusion equations, which is a technical but incremental contribution for applied mathematicians.
The paper develops a finite difference method for a singularly perturbed linear parabolic system with distinct small parameters, achieving first-order convergence in time and essentially second-order convergence in space, uniformly with respect to all parameters.
A singularly perturbed linear system of second order partial differential equations of parabolic reaction-diffusion type with given initial and 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 are first order convergent in time and essentially second order convergent in the space variable uniformly with respect to all of the parameters.