scholarly journals AN OVERLAPPING SCHWARZ METHOD FOR SINGULARLY PERTURBED FOURTH-ORDER CONVECTION-DIFFUSION TYPE

2020 ◽  
Vol 25 (4) ◽  
pp. 661-679
Author(s):  
J. Christy Roja ◽  
Ayyadurai Tamilselvan
Keyword(s):  
Difference Scheme ◽  
Fourth Order ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Diffusion Type ◽  
Convection Diffusion ◽  
Layer Region ◽  
Original Domain ◽  
Central Finite Difference ◽  
Overlapping Schwarz

In this paper, we have constructed an iterative numerical method based on an overlapping Schwarz procedure with uniform mesh for singularly perturbed fourth-order of convection-diffusion type. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region we use the central finite difference scheme on a uniform mesh while on the non-layer region we use the mid-point difference scheme on a uniform mesh. It is shown that the method produces numerical approximations which converge in the maximum norm to the exact solution. We prove that, when appropriate subdomains are used the method produces convergence of almost second-order. Furthermore, it is shown that, two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results.

scholarly journals A NUMERICAL SCHEME FOR SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS OF CONVECTION-DIFFUSION TYPE ON AN ADAPTIVE GRID

2018 ◽  
Vol 23 (4) ◽  
pp. 686-698 ◽  
Author(s):  
ramod Chakravarthy Podila ◽  
Trun Gupta ◽  
Nageshwar Rao
Keyword(s):  
Delay Differential Equations ◽  
Numerical Scheme ◽  
Adaptive Mesh ◽  
Singularly Perturbed ◽  
Adaptive Meshes ◽  
Diffusion Type ◽  
Convection Diffusion ◽  
Computational Work ◽  
Delay Differential ◽  
Central Finite Difference

In this paper, an adaptive mesh strategy is presented for solving singularly perturbed delay differential equation of convection-diffusion type using second order central finite difference scheme. Layer adaptive meshes are generated via an entropy production operator. The details of the location and width of the layer is not required in the proposed method unlike the popular layer adaptive meshes mainly by Bakhvalov and Shishkin. An extensive amount of computational work has been carried out to demonstrate the applicability of the proposed method.

scholarly journals A Higher-Order Finite Difference Scheme for Singularly Perturbed Parabolic Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Shifang Tian ◽  
Xiaowei Liu ◽  
Ran An
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Diffusion Problem ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Third Order ◽  
Order Accuracy ◽  
Convection Diffusion ◽  
Backward Euler

In this paper, we deal with a singularly perturbed parabolic convection-diffusion problem. Shishkin mesh and a hybrid third-order finite difference scheme are adopted for the spatial discretization. Uniform mesh and the backward Euler scheme are used for the temporal discretization. Furthermore, a preconditioning approach is also used to ensure uniform convergence. Numerical experiments show that the method is first-order accuracy in time and almost third-order accuracy in space.

A parameter-uniform finite difference scheme for singularly perturbed parabolic problem with two small parameters

2021 ◽  
Author(s):  
Tesfaye Aga Bullo ◽  
Guy Aymard Degla ◽  
Gemechis File Duressa
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Variable Parameter ◽  
Accurate Solution ◽  
Singularly Perturbed ◽  
Finite Difference Approximation ◽  
Parabolic Problems ◽  
Difference Approximation ◽  
Uniform Mesh

A parameter-uniform finite difference scheme is constructed and analyzed for solving singularly perturbed parabolic problems with two parameters. The solution involves boundary layers at both the left and right ends of the solution domain. A numerical algorithm is formulated based on uniform mesh finite difference approximation for time variable and appropriate piecewise uniform mesh for the spatial variable. Parameter-uniform error bounds are established for both theoretical and experimental results and observed that the scheme is second-order convergent. Furthermore, the present method produces a more accurate solution than some methods existing in the literature.   

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