Collocation method using artificial viscosity for solving stiff singularly perturbed turning point problem having twin boundary layers

Kadalbajoo, Mohan K.; Arora, Puneet; Gupta, Vikas

doi:10.1016/j.camwa.2011.01.028

Cited by 27 publications

(19 citation statements)

References 13 publications

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“…Thus with almost same computational effort, proposed finite difference scheme gives more accuracy and rapid convergence then the finite difference scheme [20]. We also compare proposed finite difference scheme with the spline based numerical methods [11,12,13] numerically for both the Examples 1-2 and found that present scheme produce lesser pointwise errors and larger order of convergence than the spline based numerical methods [11,12,13]. Furthermore, one can see, Example 1 is analogous to the Testproblem 1 in [4] and our results are comparable to those extrapolation results in [4] as both numerical schemes are of almost second order convergence O(N −2 ln(N ) 2 ) under the common assumption ε ≤ CN −1 for a given number of mesh points N .…”

Section: Numerical Results and Discussionmentioning

confidence: 93%

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

Gupta

Kadalbajoo

Dubey

2018

International Journal of Computer Mathematics

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In this paper, a parameter-uniform fitted mesh finite difference scheme is constructed and analyzed for a class of singularly perturbed interior turning point problems. The solution of this class of turning point problem possess two outflow exponential boundary layers. Parameter-explicit theoretical bounds on the derivatives of the analytical solution are given, which are used in the error analysis of the proposed scheme. The problem is discretized by a hybrid finite difference scheme comprises of midpoint-upwind and central difference operator on an appropriate piecewise-uniform fitted mesh. An error analysis has been carried out for the proposed scheme by splitting the solution into regular and singular components and the method has been shown second order uniform convergent except for a logarithmic factor with respect to the singular perturbation parameter. Some relevant numerical examples are also illustrated to verify computationally the theoretical aspects.Numerical experiments show that the proposed method gives competitive results in comparison to those of other methods exist in the literature.

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Section: Numerical Results and Discussionmentioning

confidence: 93%

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

Gupta

Kadalbajoo

Dubey

2018

International Journal of Computer Mathematics

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“…Taking s i = −1 + 2i n+1 , n = 100,the maximum absolute errors using the present method (PM) are compared with [2] and [8] in Table 4. The absolute errors obtained using the present method for ε = 2 −12 , n = 100 are shown in Fig.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Under the above assumptions the given turning point problem has a solution with two boundary layers at x = ±1 [2]. The rest of the paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

“…Natesan et al [1] proposed a parameter uniform numerical method for singularly perturbed turning point problems exhibiting boundary layers. Kadalbajoo et al [2] solved stiff singularly perturbed turning point problem having twin boundary layers by a collocation method. Rai and Sharma [3][4][5] developed some numerical methods for solving singularly perturbed differential-difference equations with turning points.…”

Section: Introductionmentioning

confidence: 99%

“…Such behavior results in some major computational difficulties in the numerical treatment of singularly perturbed problems. Standard finite difference or finite element methods used on uniform meshes can not obtain accurate approximate solutions and in recent years a large number of special-purpose methods have been proposed [1][2][3][4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

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A new numerical method for singularly perturbed turning point problems with two boundary layers based on reproducing kernel method

Geng

Qian

2016

Calcolo

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In this paper, a simple numerical method is proposed for solving singularly perturbed boundary layers problems exhibiting twin boundary layers. The method avoids the choice of fitted meshes. Firstly the original problem is transformed into a new boundary value problem whose solution does not change rapidly by a proper variable transformation; then the transformed problem is solved by using the reproducing kernel method. Two numerical examples are given to show the effectiveness of the present method.

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Solving Singular Perturbation Problems by B-Spline and Artificial Viscosity Method

Chang

Yang

Zhao

2011

Information Computing and Applications

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Collocation method using artificial viscosity for solving stiff singularly perturbed turning point problem having twin boundary layers

Cited by 27 publications

References 13 publications

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

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

A new numerical method for singularly perturbed turning point problems with two boundary layers based on reproducing kernel method

Solving Singular Perturbation Problems by B-Spline and Artificial Viscosity Method

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