A hybrid difference scheme for a singularly perturbed convection-diffusion problem with discontinuous convection coefficient

Cen, Zhongdi

doi:10.1016/j.amc.2004.08.051

Cited by 39 publications

(38 citation statements)

References 2 publications

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“…Then, following the arguments given in [1], one can deduce that the truncation error satisfies the following estimates…”

Section: Lemmamentioning

confidence: 99%

“…Now, using the assumption ε ≤ 2 a N −1 and invoking Theorem 1 and Lemma 1, by means of the barrier function approach as given in [1], one can obtain the desired error estimate. Hence, this completes the proof.…”

Section: Error In the Outer Regionmentioning

confidence: 99%

“…Afterwards in Section 5, we prove the main convergence result related to the ε-uniform error bound of the proposed numerical scheme. Finally, we carry out the extensive numerical experiments in Section 6 to validate the theoretical results and also to demonstrate the efficiency and accuracy of the proposed scheme, we compare the numerical results obtained by the proposed hybrid scheme with the hybrid scheme developed in [1]. We end up this section by stating observations about the newly proposed scheme with concluding remarks.…”

mentioning

confidence: 99%

“…However, it is worth mentioning that the hybrid numerical scheme proposed by Cen in [1], is an well-known fitted mesh method for solving singularly perturbed BVPs of the form (1.1)-(1.2) with discontinuous convection coefficient and the method is almost second-order accurate throughout the domain [0, 1] provided the perturbation parameter ε satisfies ε N −1 , otherwise the method is at worst first-order uniformly convergent with respect to ε in the discrete supremum norm (see the detailed discussion in Section 6). Therefore, it is quite natural to ask whether one can design a hybrid scheme which attains an improvement with respect to the ε-uniform order of convergence, compared to the above mentioned hybrid scheme.…”

mentioning

confidence: 99%

“…Over the last few years, several researchers developed the fitted mesh methods for solving singularly perturbed problems with non-smooth data, one can refer the articles [1,2,4,10,12] for the stationary case and [7,8,9] for the nonstationary case. However, it is worth mentioning that the hybrid numerical scheme proposed by Cen in [1], is an well-known fitted mesh method for solving singularly perturbed BVPs of the form (1.1)-(1.2) with discontinuous convection coefficient and the method is almost second-order accurate throughout the domain [0, 1] provided the perturbation parameter ε satisfies ε N −1 , otherwise the method is at worst first-order uniformly convergent with respect to ε in the discrete supremum norm (see the detailed discussion in Section 6).…”

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confidence: 99%

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Parameter-Uniform Improved Hybrid Numerical Scheme for Singularly Perturbed Problems With Interior Layers

Mukherjee

2018

Mathematical Modelling and Analysis

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Abstract. In this paper, we consider a class of singularly perturbed convectiondiffusion boundary-value problems with discontinuous convection coefficient which often occur as mathematical models for analyzing shock wave phenomena in gas dynamics. In general, interior layers appear in the solutions of this class of problems and this gives rise to difficulty while solving such problems using the classical numerical methods (standard central difference or standard upwind scheme) on uniform meshes when the perturbation parameter ε is small. To achieve better numerical approximation in solving this class of problems, we propose a new hybrid scheme utilizing a layer-resolving piecewise-uniform Shishkin mesh and the method is shown to be ε-uniformly stable. In addition to this, it is proved that the proposed numerical scheme is almost second-order uniformly convergent in the discrete supremum norm with respect to the parameter ε. Finally, extensive numerical experiments are conducted to support the theoretical results. Further, the numerical results obtained by the newly proposed scheme are also compared with the hybrid scheme developed in the paper [Z.Cen, Appl. Math. Comput., 169(1): 689-699, 2005]. It shows that the current hybrid scheme exhibits a significant improvement over the hybrid scheme developed by Cen, in terms of the parameter-uniform order of convergence.Keywords: singularly perturbed boundary-value problem, interior layer, numerical scheme, piecewise-uniform Shishkin mesh, uniform convergence.

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“…Then, following the arguments given in [1], one can deduce that the truncation error satisfies the following estimates…”

Section: Lemmamentioning

confidence: 99%

Section: Error In the Outer Regionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Parameter-Uniform Improved Hybrid Numerical Scheme for Singularly Perturbed Problems With Interior Layers

Mukherjee

2018

Mathematical Modelling and Analysis

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Robust computational method for singularly perturbed system of parabolic convection‐diffusion problems with interior layers

Natesan

Singh

2021

Comp and Math Methods

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In this article, we present the convergence analysis of an upwind finite difference scheme for singularly perturbed system of parabolic convection-diffusion initial-boundary-value problems with discontinuous convection coefficient and source term. The proposed numerical scheme is constructed by using the implicit-Euler scheme for the time derivative on the uniform mesh, and the upwind finite difference scheme for the spatial derivatives on a layer-resolving piecewise-uniform Shishkin mesh. It is shown that the numerical solution obtained by the proposed scheme converges uniformly with respect to the perturbation parameter. The proposed numerical 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.

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Numerical treatment of two‐parameter singularly perturbed parabolic convection diffusion problems with non‐smooth data

Chandru

Das

Ramos

2018

Math Methods in App Sciences

112

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In the present work, we consider a parabolic convection‐diffusion‐reaction problem where the diffusion and convection terms are multiplied by two small parameters, respectively. In addition, we assume that the convection coefficient and the source term of the partial differential equation have a jump discontinuity. The presence of perturbation parameters leads to the boundary and interior layers phenomena whose appropriate numerical approximation is the main goal of this paper. We have developed a uniform numerical method, which converges almost linearly in space and time on a piecewise uniform space adaptive Shishkin‐type mesh and uniform mesh in time. Error tables based on several examples show the convergence of the numerical solutions. In addition, several numerical simulations are presented to show the effectiveness of resolving layer behavior and their locations.

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A hybrid difference scheme for a singularly perturbed convection-diffusion problem with discontinuous convection coefficient

Cited by 39 publications

References 2 publications

Parameter-Uniform Improved Hybrid Numerical Scheme for Singularly Perturbed Problems With Interior Layers

Parameter-Uniform Improved Hybrid Numerical Scheme for Singularly Perturbed Problems With Interior Layers

Robust computational method for singularly perturbed system of parabolic convection‐diffusion problems with interior layers

Numerical treatment of two‐parameter singularly perturbed parabolic convection diffusion problems with non‐smooth data

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