A higher order scheme for singularly perturbed delay parabolic turning point problem

Abstract

Purpose The purpose of this study is to construct and analyze a parameter uniform higher-order scheme for singularly perturbed delay parabolic problem (SPDPP) of convection-diffusion type with a multiple interior turning point. Design/methodology/approach The authors construct a higher-order numerical method comprised of a hybrid scheme on a generalized Shishkin mesh in space variable and the implicit Euler method on a uniform mesh in the time variable. The hybrid scheme is a combination of simple upwind scheme and the central difference scheme. Findings The proposed method has a convergence rate of order O(N−2L2+Δt). Further, Richardson extrapolation is used to obtain convergence rate of order two in the time variable. The hybrid scheme accompanied with extrapolation is second-order convergent in time and almost second-order convergent in space up to a logarithmic factor. Originality/value A class of SPDPPs of convection-diffusion type with a multiple interior turning point is studied in this paper. The exact solution of the considered class of problems exhibit two exponential boundary layers. The theoretical results are supported via conducting numerical experiments. The results obtained using the proposed scheme are also compared with the simple upwind scheme.

A second-order numerical method for pseudo-parabolic equations having both layer behavior and delay argument

“…al [41] have designed a numerical discretization using fitted operator finite difference method on spatially direction and Crank Nicolson finite difference approach on time direction. S. Yadav and P. Rai [62] have constructed a higher-order difference method consisting of hybrid scheme on Shishkin mesh and implicit Euler method on a uniform mesh to examine singularly perturbed delay parabolic turning point problems of convection-diffusion type. Authors in [10,12] have provided the standard finite difference scheme on piecewise uniform fitted mesh to analyze singularly perturbed delay parabolic initial-boundary value problems.…”

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)).…”