Numerical scheme for partial differential equations involving small diffusion term with non-local boundary conditions

In this paper, we present a novel numerical method designed for partial differential equations with small diffusion terms and non-local boundary conditions. Our approach involves the discretization of the temporal derivative through the implicit Euler method, while the spatial derivatives are discretized using the central difference method, resulting in a 2nd-order convergence rate, initially. To further enhance the order of convergence and accuracy, we employ the extrapolation method. We extend our analysis to prove that the proposed numerical scheme achieves epsilon-uniform convergence and demonstrates a remarkable maximum convergence rate of up to 4th-order. To validate the theoretical findings, we conduct a series of numerical experiments utilizing our developed technique, providing quantitative results that affirm the effectiveness of our approach in handling singularly perturbed partial differential equations characterized by non-local boundary conditions.