Parameter‐uniform numerical method for singularly perturbed 2‐D parabolic convection–diffusion problem with interior layers

• Published in: Mathematical methods in the applied sciences Vol. 45; no. 5; pp. 3039 - 3057
• Main Authors: Majumdar, Anirban, Natesan, Srinivasan
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 30.03.2022
• Subjects: Convection; Convergence; Diffusion layers; finite difference scheme; interior layer; Numerical methods; piecewise‐uniform meshes; Singular perturbation methods; singularly perturbed 2‐D parabolic convection–diffusion problem; uniform convergence
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.7975

In this article, we devise a uniformly convergent numerical scheme for solving singularly perturbed two‐dimensional parabolic convection–diffusion problem with non‐smooth convection coefficients and source term. The solution of this kind of problem typically exhibits interior layers due to the discontinuity of convection coefficients and source term. To capture the interior layers, the piecewise‐uniform mesh is used in the spatial directions and the uniform mesh is considered in temporal direction. To discretize the temporal and spatial derivatives, we apply an alternating direction method and upwind method, respectively. Theoretically, we prove that the proposed method is ε‐uniformly convergent. Numerical results are presented to demonstrate the theoretical estimates.