An almost fourth order uniformly convergent domain decomposition method for a coupled system of singularly perturbed reaction–diffusion equations

Rao, S. Chandra Sekhara; Kumar, Sunil

doi:10.1016/j.cam.2011.01.047

Cited by 22 publications

(3 citation statements)

References 14 publications

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“…The extended cubic B-spline is considered in [15]. A domain decomposition method is considered in [16,17]. The authors in [18,19] proposed hybrid scheme on both Shishkin and Bakhvalov meshes.…”

Section: Introductionmentioning

confidence: 99%

Fitted computational method for singularly perturbed convection-diffusion equation with time delay

Tesfaye,

Duressa,

Woldaregay

et al. 2023

Front. Appl. Math. Stat.

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A uniformly convergent numerical scheme is proposed to solve a singularly perturbed convection-diffusion problem with a large time delay. The diffusion term of the problem is multiplied by a perturbation parameter, ε. For a small ε, the problem exhibits a boundary layer, which makes it challenging to solve it analytically or using standard numerical methods. As a result, the backward Euler scheme is applied in the temporal direction. Non-symmetric finite difference schemes are applied for approximating the first-order derivative terms, and a higher-order finite difference method is applied for approximating the second-order derivative term. Furthermore, an exponential fitting factor is computed and induced in the difference scheme to handle the effect of the small parameter. Using the discrete maximum principle, the stability of the scheme is examined and analyzed. The developed scheme is parameter-uniform with a linear order of convergence in both space and time. To examine the accuracy of the method, two model examples are considered. Further, the boundary layer behavior of the solutions is given graphically.

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Section: Introductionmentioning

confidence: 99%

Fitted computational method for singularly perturbed convection-diffusion equation with time delay

Tesfaye,

Duressa,

Woldaregay

et al. 2023

Front. Appl. Math. Stat.

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“…In [20], an analysis of overlapping domain decomposition methods for singularly perturbed reaction-diffusion problems with distinct small positive parameters is presented. The authors of [20] found a flaw in the analysis of domain decomposition methods explored in [6,13,18]. The authors observation is that the constant C is not independent of the iteration number k and it is growing at each induction step in their proof of [6,13,18].…”

mentioning

confidence: 99%

“…The authors of [20] found a flaw in the analysis of domain decomposition methods explored in [6,13,18]. The authors observation is that the constant C is not independent of the iteration number k and it is growing at each induction step in their proof of [6,13,18]. But in [20] the authors have presented an alternate analysis of overlapping domain decomposition methods for singularly perturbed reaction-diffusion problems with two parameters and problems in [18].…”

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confidence: 99%

An Overlapping Schwarz Method for Singularly Perturbed Fourth-Order Convection-Diffusion Type

Roja

Tamilselvan

2020

Mathematical Modelling and Analysis

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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.

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An analysis of overlapping Schwarz method for a weakly coupled system of singularly perturbed convection‐diffusion equations

Roja¹,

Tamilselvan

Geetha

2019

Numerical Methods in Fluids

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In this article, we have developed an overlapping Schwarz method for a weakly coupled system of convection-diffusion equations. 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, whereas on the nonlayer region, we use the mid-point difference scheme on a uniform mesh. It is shown that the numerical approximations converge in the maximum norm to the exact solution. We have proved that, when appropriate subdomains are used, the method produces almost second-order convergence. Furthermore, it is shown that two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results. The main advantage of this method used with the proposed scheme is that it reduces iteration counts very much and easily identifies in which iteration the Schwarz iterate terminates. K E Y W O R D Sconvection-diffusion equations, hybrid difference scheme, Schwarz method, singularly perturbed problems, weakly coupled system

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An almost fourth order uniformly convergent domain decomposition method for a coupled system of singularly perturbed reaction–diffusion equations

Cited by 22 publications

References 14 publications

Fitted computational method for singularly perturbed convection-diffusion equation with time delay

Fitted computational method for singularly perturbed convection-diffusion equation with time delay

An Overlapping Schwarz Method for Singularly Perturbed Fourth-Order Convection-Diffusion Type

An analysis of overlapping Schwarz method for a weakly coupled system of singularly perturbed convection‐diffusion equations

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