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

• Published in: Journal of applied mathematics & computing Vol. 69; no. 6; pp. 4307 - 4331
• Main Authors: Bala, Shree, Govindarao, L., Das, A., Majumdar, A.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2023; Springer Nature B.V
• Subjects: Boundary conditions; Computational Mathematics and Numerical Analysis; Convergence; Discretization; Extrapolation; Mathematical analysis; Mathematical and Computational Engineering; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical methods; Original Research; Partial differential equations; Quantitative analysis; Theory of Computation; Central difference scheme; 65M12; 65M06; Partial differential equation; Non-local boundary condition; Boundary layer
• ISSN: 1598-5865; 1865-2085
• DOI: 10.1007/s12190-023-01927-y

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 ε -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.