Uniformly Convergent Hybrid Numerical Method for Singularly Perturbed Delay Convection-Diffusion Problems

Abstract

This paper deals with numerical treatment of nonstationary singularly perturbed delay convection-diffusion problems. The solution of the considered problem exhibits boundary layer on the right side of the spatial domain. To approximate the term with the delay, Taylor’s series approximation is used. The resulting time-dependent singularly perturbed convection-diffusion problems are solved using Crank-Nicolson method for temporal discretization and hybrid method for spatial discretization. The hybrid method is designed using mid-point upwind in regular region with central finite difference in boundary layer region on piecewise uniform Shishkin mesh. Numerical examples are used to validate the theoretical findings and analysis of the proposed scheme. The present method gives accurate and nonoscillatory solutions in regular and boundary layer regions of the solution domain. The stability and the uniform convergence of the scheme are proved. The scheme converges uniformly with almost second-order rate of convergence.

Second-order robust finite difference method for singularly perturbed Burgers' equation

“…Therefore, Eq. (3.10), is convergent scheme by Lax's equivalence theorem, one can refer to the details in [1], [2], [3], [17,18,19,20,21]. Further, the local truncation error is bounded with confirmation of its norm as…”

“…Also, these problems have different properties and modeled problems depending on the dimensions, number of parameters involved, natures of the models, etc. For instance, some of them are singularly perturbed Burgers' equations, Burger-Huxley equations, Burger-Fisher equations, and so on, see the reference in [1,2,3,4,5,6,7,8,9]. Singularly perturbed Burgers' equation first introduced by Bateman in 1915 [9] is used in the modeling of the motion of the viscous fluid.…”

“…Also, these problems have different properties and modeled problems depending on the dimensions, number of parameters involved, natures of the models, etc. For instance, some of them are singularly perturbed Burgers' equations, Burger-Huxley equations, Burger-Fisher equations, and so on, see the reference in [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] .…”

Fitted Tension Spline Method for Singularly Perturbed Time Delay Reaction Diffusion Problems

“…Woldaregay et al [3,[12][13][14][15][16] developed an exponentially tted numerical method for solving di erent singularly perturbed di erential-di erence equations. In [14,17], they proposed the nonstandard nite di erence techniques and the tted mesh methods, respectively. In [18,19], an initial value technique with an exponential tting factor is applied to solve convection-dominated delay differential equations.…”

Numerical Treatment on Parabolic Singularly Perturbed Differential Difference Equation via Fitted Operator Scheme

“…Clavero and Gracia [4] studied singularly perturbed time-dependent problem of reaction-diffusion type using Richardson extrapolation technique. Woldaregay and Duressa [5] considered a numerical method for both small time delay and large time delay singularly perturbed boundary value problem. Kumar and Kumari [6] show that the influence of a small delay on the solution is extremely sensitive and that a small change in the delay can have a significant impact on the solution.…”