An exponentially fitted non-polynomial spline approach for singularly perturbed parabolic reaction-diffusion equation with large negative shift

This article addresses an exponentially fitted numerical scheme for singularly perturbed partial differential equation (SPPDE) involving a large negative shift. Due to the perturbation parameter, the solution exhibits a rapidly changing twin boundary layers, and the large shift term causes an interior layer. Such rapidly varying behavior of the layers induce challenges in computing the exact solution. On the other hand, standard numerical methods do not yield satisfactory results, as they do not consider the layer behaviors of the solution. To overcome these challenges, we formulate a parameter uniform numerical scheme by discretizing the problem using the Crank-Nicolson technique in the time variable, and an exponentially fitted non-polynomial spline approach in the space variable on uniform grid points. We establish and prove the stability and uniform error estimate for the formulated numerical scheme. To test the applicability and validity of the formulated numerical scheme, we conduct numerical experiments and confirm with the theoretical results. The uniform error estimate and the numerical experiments show that the present method converges uniformly with second-order convergence rate in time and in the spatial directions.