A graded mesh refinement approach for boundary layer originated singularly perturbed time‐delayed parabolic convection diffusion problems

• Published in: Mathematical methods in the applied sciences Vol. 44; no. 16; pp. 12332 - 12350
• Main Authors: Kumar, Kamalesh, Podila, Pramod Chakravarthy, Das, Pratibhamoy, Ramos, Higinio
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 15.11.2021
• Subjects: adaptive mesh algorithm; algorithmic complexity; Algorithms; boundary layer phenomena; Boundary layers; Diffusion layers; entropy; graded mesh; Grid refinement (mathematics); Mesh generation; parabolic convection–diffusion problems; Parabolic differential equations; Parameters; Partial differential equations; Perturbation; time delay
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.7358

In this work, we consider a graded mesh refinement algorithm for solving time‐delayed parabolic partial differential equations with a small diffusion parameter. The presence of this parameter leads to boundary layer phenomena. These problems are also known as singularly perturbed problems. For these problems, it is well‐known that one cannot achieve a convergent solution to maintain the boundary layer dynamics, on a fixed number of uniform meshes irrespective of the arbitrary magnitude of perturbation parameter. Here, we consider an adaptive graded mesh generation algorithm, which is based on an entropy function in conjunction with the classical difference schemes, to resolve the layer behavior. The advantage of the present algorithm is that it does not require to have any information about the location of the layer. Several examples are presented to show the high performance of the proposed algorithm.