Lolugu Govindarao and Jugal Mohapatra
The purpose of this paper is to provide an efficient and robust second-order monotone hybrid scheme for singularly perturbed delay parabolic convection-diffusion initial boundary…
Abstract
Purpose
The purpose of this paper is to provide an efficient and robust second-order monotone hybrid scheme for singularly perturbed delay parabolic convection-diffusion initial boundary value problem.
Design/methodology/approach
The delay parabolic problem is solved numerically by a finite difference scheme consists of implicit Euler scheme for the time derivative and a monotone hybrid scheme with variable weights for the spatial derivative. The domain is discretized in the temporal direction using uniform mesh while the spatial direction is discretized using three types of non-uniform meshes mainly the standard Shishkin mesh, the Bakhvalov–Shishkin mesh and the Gartland Shishkin mesh.
Findings
The proposed scheme is shown to be a parameter-uniform convergent scheme, which is second-order convergent and optimal for the case. Also, the authors used the Thomas algorithm approach for the computational purposes, which took less time for the computation, and hence, more efficient than the other methods used in literature.
Originality/value
A singularly perturbed delay parabolic convection-diffusion initial boundary value problem is considered. The solution of the problem possesses a regular boundary layer. The authors solve this problem numerically using a monotone hybrid scheme. The error analysis is carried out. It is shown to be parameter-uniform convergent and is of second-order accurate. Numerical results are shown to verify the theoretical estimates.
Details
Keywords
Lolugu Govindarao and Jugal Mohapatra
The purpose of this paper is to provide a robust second-order numerical scheme for singularly perturbed delay parabolic convection–diffusion initial boundary value problem.
Abstract
Purpose
The purpose of this paper is to provide a robust second-order numerical scheme for singularly perturbed delay parabolic convection–diffusion initial boundary value problem.
Design/methodology/approach
For the parabolic convection-diffusion initial boundary value problem, the authors solve the problem numerically by discretizing the domain in the spatial direction using the Shishkin-type meshes (standard Shishkin mesh, Bakhvalov–Shishkin mesh) and in temporal direction using the uniform mesh. The time derivative is discretized by the implicit-trapezoidal scheme, and the spatial derivatives are discretized by the hybrid scheme, which is a combination of the midpoint upwind scheme and central difference scheme.
Findings
The authors find a parameter-uniform convergent scheme which is of second-order accurate globally with respect to space and time for the singularly perturbed delay parabolic convection–diffusion initial boundary value problem. Also, the Thomas algorithm is used which takes much less computational time.
Originality/value
A singularly perturbed delay parabolic convection–diffusion initial boundary value problem is considered. The solution of the problem possesses a regular boundary layer. The authors solve this problem numerically using a hybrid scheme. The method is parameter-uniform convergent and is of second order accurate globally with respect to space and time. Numerical results are carried out to verify the theoretical estimates.