A parameter uniform numerical method on a Bakhvalov type mesh for singularly perturbed degenerate parabolic convection–diffusion problems

• Published in: Journal of applied mathematics & computing Vol. 70; no. 6; pp. 5645 - 5668
• Main Authors: Kumar, Shashikant, Kumar, Sunil, Ramos, Higinio, Kuldeep
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2024; Springer Nature B.V
• Subjects: Boundary layers; Computational Mathematics and Numerical Analysis; Convection; Convergence; Diffusion barriers; Diffusion layers; Error analysis; Mathematical and Computational Engineering; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical methods; Original Research; Singular perturbation; Theory of Computation; Truncation errors; Uniform convergence; 65M15; Singular perturbation; 65M06; Bakhvalov mesh; 65M55; 65M12; Upwind scheme; Degenerate parabolic problem
• ISSN: 1598-5865; 1865-2085
• DOI: 10.1007/s12190-024-02178-1

We are focused on the numerical treatment of a singularly perturbed degenerate parabolic convection–diffusion problem that exhibits a parabolic boundary layer. The discretization and analysis of the problem are done in two steps. In the first step, we discretize in time and prove its uniform convergence using an auxiliary problem. In the second step, we discretize in space using an upwind scheme on a Bakhvalov-type mesh and prove its uniform convergence using the truncation error and barrier function approach, wherein several bounds derived for the mesh step sizes are used. Numerical results for a couple of examples are presented to support the theoretical bounds derived in the paper.