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Abhishek Das

Bio: Abhishek Das is an academic researcher from Institute of Chartered Financial Analysts of India. The author has contributed to research in topics: Discretization & Convection–diffusion equation. The author has an hindex of 4, co-authored 7 publications receiving 62 citations. Previous affiliations of Abhishek Das include ICFAI University, Tripura & Indian Institute of Technology Guwahati.

Papers
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TL;DR: This article studies the numerical solution of singularly perturbed delay parabolic convection-diffusion initial-boundary-value problems and uses the piecewise-uniform Shishkin mesh for the discretization of the domain in the spatial direction, and uniform mesh in the temporal direction.

46 citations

Journal ArticleDOI
TL;DR: A second-order uniformly convergent numerical method for singularly perturbed delay parabolic convection-diffusion equation having a regular boundary layer and the implementation of Richardson extrapolation technique enhanced the order of convergence.
Abstract: This article proposes a second-order uniformly convergent numerical method for singularly perturbed delay parabolic convection-diffusion equation having a regular boundary layer. To handle this lay...

25 citations

Journal ArticleDOI
TL;DR: In this paper, a higher-order parameter uniformly convergent method for a singularly perturbed delay parabolic reaction-diffusion initial-boundary-value problem is presented.
Abstract: This article presents a higher-order parameter uniformly convergent method for a singularly perturbed delay parabolic reaction–diffusion initial-boundary-value problem. For the discretization of the time derivative, we use the implicit Euler scheme on the uniform mesh and for the spatial discretization, we use the central difference scheme on the Shishkin mesh, which provides a second-order convergence rate. To enhance the order of convergence, we apply the Richardson extrapolation technique. We prove that the proposed method converges uniformly with respect to the perturbation parameter and also attains almost fourth-order convergence rate. Finally, to support the theoretical results, we present some numerical experiments by using the proposed method.

22 citations

Journal ArticleDOI
TL;DR: A second-order uniformly convergent numerical method for a singularly perturbed 2D parabolic convection–diffusion initial–boundary-value problem and uses the Richardson extrapolation technique to enhance the order of convergence.
Abstract: In this article, we propose a second-order uniformly convergent numerical method for a singularly perturbed 2D parabolic convection–diffusion initial–boundary-value problem. First, we use a fractional-step method to discretize the time derivative of the continuous problem on uniform mesh in the temporal direction, which gives a set of two 1D problems. Then, we use the classical finite difference scheme to discretize those 1D problems on a special mesh, which results almost first-order convergence, i.e., O ( N − 1 + β ln N + Δ t ) . To enhance the order of convergence to O ( N − 2 + β ln 2 N + Δ t 2 ) , we use the Richardson extrapolation technique. In support of the theoretical results, numerical experiments are performed by employing the proposed technique.

12 citations

Journal ArticleDOI
TL;DR: In this article, the numerical solution of a singularly perturbed 2D delay parabolic convection-diffusion problem is studied, where the authors discretize the domain with a uniform mesh in the temporal direction and a special meshes in the spatial direction.
Abstract: In this article, we study the numerical solution of a singularly perturbed 2D delay parabolic convection–diffusion problem. First, we discretize the domain with a uniform mesh in the temporal direction and a special mesh in the spatial directions. The numerical scheme used to discretize the continuous problem, consists of the implicit-Euler scheme for the time derivative and the classical upwind scheme for the spatial derivatives. Stability analysis is carried out, and parameter-uniform error estimates are derived. The proposed scheme is of almost first-order (up to a logarithmic factor) in space and first-order in time. Numerical examples are carried out to verify the theoretical results.

6 citations


Cited by
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Journal ArticleDOI
TL;DR: This paper studies the numerical solutions of singularly perturbed parabolic convection–diffusion problems with a delay in time, and proves that the proposed scheme is -uniform convergence of first-order in time and first- order up to a logarithmic factor in space.
Abstract: This paper studies the numerical solutions of singularly perturbed parabolic convection–diffusion problems with a delay in time. We divide the domain using a piecewise uniform adaptive mesh in the spatial direction and a uniform mesh in the temporal direction. Further, we discretize the time derivative by the backward-Euler scheme and the spatial derivatives by the upwind finite difference scheme. We obtain the maximum principle and carry out the stability analysis. Then we prove that the proposed scheme is -uniform convergence of first-order in time and first-order up to a logarithmic factor in space. Numerical results are carried out to verify the theoretical results.

43 citations

Journal ArticleDOI
TL;DR: An adaptive graded mesh generation algorithm, which is based on an entropy function in conjunction with the classical difference schemes, to resolve the layer behavior and several examples are presented to show the high performance of the proposed algorithm.

36 citations

Journal ArticleDOI
TL;DR: In this paper, a numerical scheme for a class of singularly perturbed parabolic partial differential equation with the time delay on a rectangular domain in the x-t plane is constructed.
Abstract: A numerical scheme for a class of singularly perturbed parabolic partial differential equation with the time delay on a rectangular domain in the x–t plane is constructed. The presence of the perturbation parameter in the second-order space derivative gives rise to parabolic boundary layer(s) on one (or both) of the lateral side(s) of the rectangle. Thus the classical numerical methods on the uniform mesh are inadequate and fail to give good accuracy and results in large oscillations as the perturbation parameter approaches zero. To overcome this drawback a numerical method comprising the Crank–Nicolson finite difference method consisting of a midpoint upwind finite difference scheme on a fitted piecewise-uniform mesh of $$N\times M$$ elements condensing in the boundary layer region is constructed. A priori explicit bounds on the solution of the problem and its derivatives which are useful for the error analysis of the numerical method are established. To establish the parameter-uniform convergence of the proposed method an extensive amount of analysis is carried out. It is shown that the proposed difference scheme is second-order accurate in the temporal direction and the first-order (up to a logarithmic factor) accurate in the spatial direction. To validate the theoretical results, the method is applied to two test problems. The performance of the method is demonstrated by calculating the maximum absolute errors and experimental orders of convergence. Since the exact solutions of the test problems are not known, the maximum absolute errors are obtained by using double mesh principle. The numerical results show that the proposed method is simply applicable, accurate, efficient and robust.

26 citations

Journal ArticleDOI
TL;DR: A second-order uniformly convergent numerical method for singularly perturbed delay parabolic convection-diffusion equation having a regular boundary layer and the implementation of Richardson extrapolation technique enhanced the order of convergence.
Abstract: This article proposes a second-order uniformly convergent numerical method for singularly perturbed delay parabolic convection-diffusion equation having a regular boundary layer. To handle this lay...

25 citations

Journal ArticleDOI
TL;DR: In this paper, a higher-order parameter uniformly convergent method for a singularly perturbed delay parabolic reaction-diffusion initial-boundary-value problem is presented.
Abstract: This article presents a higher-order parameter uniformly convergent method for a singularly perturbed delay parabolic reaction–diffusion initial-boundary-value problem. For the discretization of the time derivative, we use the implicit Euler scheme on the uniform mesh and for the spatial discretization, we use the central difference scheme on the Shishkin mesh, which provides a second-order convergence rate. To enhance the order of convergence, we apply the Richardson extrapolation technique. We prove that the proposed method converges uniformly with respect to the perturbation parameter and also attains almost fourth-order convergence rate. Finally, to support the theoretical results, we present some numerical experiments by using the proposed method.

22 citations