Numerical treatment of two‐parameter singularly perturbed parabolic convection diffusion problems with non‐smooth data

• Published in: Mathematical methods in the applied sciences Vol. 41; no. 14; pp. 5359 - 5387
• Main Authors: Chandru, M., Das, P., Ramos, H.
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 30.09.2018
• Subjects: Computer simulation; Convection; Convergence; Diffusion; Finite element method; initial‐boundary value problem; interior and boundary layer phenomena; non‐smooth data; Numerical methods; parabolic convection‐diffusion problem; parameter uniformly convergent method; Parameters; Partial differential equations; Shishkin‐type mesh; singular perturbation; two‐parameter
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.5067

In the present work, we consider a parabolic convection‐diffusion‐reaction problem where the diffusion and convection terms are multiplied by two small parameters, respectively. In addition, we assume that the convection coefficient and the source term of the partial differential equation have a jump discontinuity. The presence of perturbation parameters leads to the boundary and interior layers phenomena whose appropriate numerical approximation is the main goal of this paper. We have developed a uniform numerical method, which converges almost linearly in space and time on a piecewise uniform space adaptive Shishkin‐type mesh and uniform mesh in time. Error tables based on several examples show the convergence of the numerical solutions. In addition, several numerical simulations are presented to show the effectiveness of resolving layer behavior and their locations.