A fast higher order uniformly convergent method for semilinear parabolic singularly perturbed systems of convection diffusion type

In this paper, we study the numerical approximation of the solution of a one dimensional semilinear singularly perturbed parabolic system of convection–diffusion type. The system is characterized by distinct small positive parameters multiplying the highest order derivative term present in each equation, wherein we consider that the coupled reaction terms are nonlinear. Then, for values of the diffusion parameter that are small to an adequate extent, in general, overlapping boundary layers arise in proximity to the spatial domain’s outflow boundary. To efficiently solve the problem, the discretization relies upon an appropriate components-wise splitting linearized fractional implicit Euler approach for discretization in time alongside a special hybrid scheme on a layer adapted Shishkin mesh to discretize in space. It is proven that the proposed method achieves robust convergence of order one in time and almost two in space. The components-wise splitting approach used allows us to solve the considered problem at a significantly lower computational cost than the classical implicit Euler approach. Further, the present method produces approximations with accuracy of order almost two in space in comparison to the existing approaches having only almost order one in space. Numerical results are recorded for various examples which clearly confirm the efficiency and the uniform convergence of the numerical algorithm.