Performance of Richardson Extrapolation on Some Numerical Methods for a Singularly Perturbed Turning Point Problem Whose Solution Has Boundary Layers

Munyakazi, Justin B.; Patidar, Kailash C.

doi:10.4134/jkms.2014.51.4.679

Cited by 14 publications

(4 citation statements)

References 20 publications

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“…Kadalbajoo et al [13] also suggested B-spline collocation with artificial viscosity on uniform mesh for the same class of SPTPP (1.1)-(1.5). In [18], Munyakazi and Patidar conclude that convergence acceleration Richardson extrapolation technique on existing numerical schemes for the above class of turning point problem does not improve the rate of convergence. However, Becher and Roos [4] show that Richardson extrapolation on upwind scheme with piecewise-uniform Shishkin mesh works fine and improves the accuracy to O(N −2 ln 2 N ) under the assumption ε ≤ CN −1 .…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

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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“…A reproducing kernel method was employed for solving Equation (1.1) by Geng and Qian [13]. Munyakazi and Patidar developed a fitted-mesh finite difference method with Richardson extrapolation in [28]. For solving Equation (1.1), a fitted-operator scheme was constructed by Phaneendra et al in [34] using nonsymmetric finite differences for the first-order derivatives.…”

Section: Introductionmentioning

confidence: 99%

A Semi-Analytic Method for Solving Singularly Perturbed Twin-Layer Problems With a Turning Point

Cengizci

Kumar

Atay

2023

Mathematical Modelling and Analysis

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This computational study investigates a class of singularly perturbed second-order boundary-value problems having dual (twin) boundary layers and simple turning points. It is well-known that the classical discretization methods fail to resolve sharp gradients arising in solving singularly perturbed differential equations as the perturbation (diffusion) parameter decreases, i.e., ε → 0+. To this end, this paper proposes a semi-analytic hybrid method consisting of a numerical procedure based on finite differences and an asymptotic method called the Successive Complementary Expansion Method (SCEM) to approximate the solution of such problems. Two numerical experiments are provided to demonstrate the method’s implementation and to evaluate its computational performance. Several comparisons with the numerical results existing in the literature are also made. The numerical observations reveal that the hybrid method leads to good solution profiles and achieves this in only a few iterations.

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“…In Kadalbajoo et al a B‐spline collocation method is designed. Readers who need more information related to nonturning points time dependent singularly perturbed parabolic problems may refer to , and those who are interested in time dependent singularly perturbed parabolic problems when the turning points lead to boundary and/or interior layer(s) are referred to .…”

Section: Introductionmentioning

confidence: 99%

“…We show that the method converges uniformly of order one in both space and time variables. We also use Richardson extrapolation , as the acceleration technique to improve the accuracy and the order of convergence of the FOFDM designed up to order two in space only.…”

Section: Introductionmentioning

confidence: 99%

A fitted numerical method for parabolic turning point singularly perturbed problems with an interior layer

Munyakazi

Patidar

Sayi

2019

Numerical Methods Partial

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The objective of this paper is to construct and analyze a fitted operator finite difference method (FOFDM) for the family of time-dependent singularly perturbed parabolic convection-diffusion problems. The solution to the problems we consider exhibits an interior layer due to the presence of a turning point. We first establish sharp bounds on the solution and its derivatives. Then, we discretize the time variable using the classical Euler method. This results in a system of singularly perturbed interior layer two-point boundary value problems. We propose a FOFDM to solve the system above. Through a rigorous error analysis, we show that the scheme is uniformly convergent of order one with respect to both time and space variables. Moreover, we apply Richardson extrapolation to enhance the accuracy and the order of convergence of the proposed scheme. Numerical investigations are carried out to demonstrate the efficacy and robustness of the scheme. KEYWORDS error bounds, finite difference methods, interior layer, singularly perturbed problems, uniform convergence 1 Numer Methods Partial Differential Eq. 2019;35:2407-2422. wileyonlinelibrary.com/journal/num

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Performance of Richardson Extrapolation on Some Numerical Methods for a Singularly Perturbed Turning Point Problem Whose Solution Has Boundary Layers

Cited by 14 publications

References 20 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 Semi-Analytic Method for Solving Singularly Perturbed Twin-Layer Problems With a Turning Point

A fitted numerical method for parabolic turning point singularly perturbed problems with an interior layer

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