An Iterative Numerical Algorithm for a Strongly Coupled System of Singularly Perturbed Convection-Diffusion Problems

O’Riordan, Eugene; Stynes, Jeanne; Stynes, Martin

doi:10.1007/978-3-642-00464-3_10

Cited by 7 publications

(14 citation statements)

References 10 publications

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“…As regards the other papers that we mentioned at the beginning of Section 4.4, [29] modifies the iterative approach of [9] from reaction -diffusion to convection -diffusion systems, [30] deals only with the special case ℓ = 2 and ε 1 = ε 2 , while [1] gives a very different analysis that draws on deeper results from classical analysis but considers only upwinding on an equidistant coarse meshω c N , which -even for a scalar convection -diffusion equationcannot yield a convergence result for u − u N ∞,ω c N that is uniform in ε.…”

Section: Finite Difference Schemementioning

confidence: 98%

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Numerical Solution of Systems of Singularly Perturbed Differential Equations

Linß

Stynes

2009

Comput. Methods Appl. Math.

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A survey is given of current research into the numerical solution of timeindependent systems of second-order differential equations whose diffusion coefficients are small parameters. Such problems are in general singularly perturbed. The equations in these systems may be coupled through their reaction and/or convection terms. Only numerical methods whose accuracy is guaranteed for all values of the diffusion parameters are considered here. Some new unifying results are also presented.2000 Mathematics Subject Classification: 65-02, 65L10, 65L12, 65L50, 65M06, 65M50.

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Section: Finite Difference Schemementioning

confidence: 98%

“…Strong coupling causes interactions between boundary layers that are not fully understood at present. The main papers on this problem are [1,17,[28][29][30].…”

Section: Strongly Coupled Systemsmentioning

confidence: 99%

Numerical Solution of Systems of Singularly Perturbed Differential Equations

Linß

Stynes

2009

Comput. Methods Appl. Math.

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“…The maximum error for the case with e = 2"^^ obtained by piecewise-uniform Shishkin mesh combined with Jacobi-type iteration [12] is 3.917 x 10-^ when A^ = 1024; the maximum 1 1 £ 10"'' 3887x10"'' 0556x10"'2 .8279x10"^1 .3868x10"'^1 0-" 1.7094x10 6 .2019x10"1 .8418x10"« 9.8919x10"'' 10 "9 .1648x10 ' 1.2265x10"2 .9805x10-* 1.2304x10"'' Table 4.2 lists the maximum errors of the RSC-sinh method for e = 10-^ 10-^ 10-«, 10-^. About the selection of sinh-transform parameter ß, further computing of Example 4.2 for different ß has been implemented.…”

Section: Example 42mentioning

confidence: 99%

“…Consider the convection-diffusion problem with constant coefficients: [11,12] The exact solution of this problem is u = (1,1, + (-2,1,…”

Section: Example 42mentioning

confidence: 99%

“…Riordan et al presented a finite difference method consisting of upwinding on piecewise-uniform Shishkin meshes for the cases of m = 2 [10] and m > 2 [11]. They also used a Jacobi-type iteration to compute the solution [12]. Cenerally speaking, the problem of this type has a solution with a single boundary layer of width O(e) at a: = 0 (or a; = 1) under proper assumptions.…”

Section: Introductionmentioning

confidence: 99%

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Rational Spectral Collocation Method for a Coupled System of Singularly Perturbed Boundary Value Problems

Yang

Wang²,

2011

JCM

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A novel collocation method for a coupled system of singularly perturbed linear equations is presented. This method is based on rational spectral collocation method in barycentric form with sinh transform. By sinh transform, the original Chebyshev points are mapped into the transformed ones clustered near the singular points of the solution. The results from asymptotic analysis about the singularity solution are employed to determine the parameters in this sinh transform. Numerical experiments are carried out to demonstrate the high accuracy and efficiency of our method.Mathematics subject classification: 65L10, 65M70.

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A stabilization technique for coupled convection‐diffusion‐reaction equations

Hernández

Massart

Peerlings

et al. 2018

Numerical Meth Engineering

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Partial differential equations having diffusive, convective, and reactive terms appear in the modeling of a large variety of processes in several branches of science. Often, several species or components interact with each other, rendering strongly coupled systems of convection-diffusion-reaction equations. Exact solutions are available in extremely few cases lacking practical interest due to the simplifications made to render such equations amenable by analytical tools. Then, numerical approximation remains the best strategy for solving these problems. The properties of these systems of equations, particularly the lack of sufficient physical diffusion, cause most traditional numerical methods to fail, with the appearance of violent and nonphysical oscillations, even for the single equation case. For systems of equations, the situation is even harder due to the lack of fundamental principles guiding numerical discretization. Therefore, strategies must be developed in order to obtain physically meaningful and numerically stable approximations. Such stabilization techniques have been extensively developed for the single equation case in contrast to the multiple equations case. This paper presents a perturbation-based stabilization technique for coupled systems of one-dimensional convection-diffusion-reaction equations. Its characteristics are discussed, providing evidence of its versatility and effectiveness through a thorough assessment. KEYWORDS convection-diffusion equation, differential equations, finite element methods, stability Int J Numer Methods Eng. 2018;116:43-65.wileyonlinelibrary.com/journal/nme

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An Iterative Numerical Algorithm for a Strongly Coupled System of Singularly Perturbed Convection-Diffusion Problems

Cited by 7 publications

References 10 publications

Numerical Solution of Systems of Singularly Perturbed Differential Equations

Numerical Solution of Systems of Singularly Perturbed Differential Equations

Rational Spectral Collocation Method for a Coupled System of Singularly Perturbed Boundary Value Problems

A stabilization technique for coupled convection‐diffusion‐reaction equations

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