A parameter uniform essentially first order convergent numerical method for a parabolic singularly perturbed differential equation of reaction-diffusion type with initial and Robin boundary conditions
R. Ishwariya, J. J. H. Miller, S. Valarmathi
TL;DR
This work analyzes a class of linear parabolic singularly perturbed reaction-diffusion equations with initial and Robin boundary conditions, where the solution $u$ is smooth but exhibits parabolic boundary layers. A classical finite difference scheme on a piecewise uniform Shishkin mesh is developed to achieve parameter-uniform convergence, being first-order in time and essentially first-order in space, with error bounded by $\mathcal{O}(M^{-1} + N^{-1}\log N)$ uniformly in the perturbation parameter $\varepsilon$. The authors establish a maximum principle, decompose the solution into smooth and singular components $u=v+w$, provide detailed bounds for both components and layer functions, and prove convergence using discrete barrier arguments and a two-mesh analysis. Numerical experiments confirm the theoretical rates and illustrate the boundary-layer behavior in $\partial u/\partial x$, demonstrating the method’s practical effectiveness for singularly perturbed parabolic problems with Robin boundaries.
Abstract
In this paper, a class of linear parabolic singularly perturbed second order differential equations of reaction-diffusion type with initial and Robin boundary conditions is considered. The solution u of this equation is smooth, whereas the first derivative in the space variable exhibits parabolic boundary layers. A numerical method composed of a classical finite difference scheme on a piecewise uniform Shishkin mesh is suggested. This method is proved to be first order convergent in time and essentially first order convergent in the space variable in the maximum norm uniformly in the perturbation parameters.