Robust numerical schemes for time delayed singularly perturbed parabolic problems with discontinuous convection and source terms

Abstract

This article deals with two different numerical approaches for solving singularly perturbed parabolic problems with time delay and interior layers. In both approaches, the implicit Euler scheme is used for the time scale. In the first approach, the upwind scheme is used to deal with the spatial derivatives whereas in the second approach a hybrid scheme is used, comprising the midpoint upwind scheme and the central difference scheme at appropriate domains. Both schemes are applied on two different layer resolving meshes, namely a Shishkin mesh and a Bakhvalov–Shishkin mesh. Stability and error analysis are provided for both schemes. The comparison is made in terms of the maximum absolute errors, rates of convergence, and the computational time required. Numerical outputs are presented in the form of tables and graphs to illustrate the theoretical findings.

Numerical approximation of parabolic singularly perturbed problems with large spatial delay and turning point

“…Several numerical approaches have been developed by many researchers for solving parabolic SP-BVPs with time delay (Ansari et al ., 2007; Gowrisankar and Natesan, 2014; Kaushik et al ., 2010; Govindarao and Mohapatra, 2019; Yadav and Rai, 2021; Kumar et al ., 2021; Priyadarshana et al ., 2024). But the corresponding class of parabolic SPPs of convection–diffusion type with spatial delay do not receive much attention and have been studied only for small values of the delay argument (see (Kumar and Kadalbajoo, 2011; Ramesh and Kadalbajoo, 2008; Rathish Kumar and Kumar, 2015)).…”