The Numerical Solution of Singular Perturbations of Boundary Value Problems

Dorr, Fred W.

doi:10.1137/0707021

Cited by 34 publications

(14 citation statements)

References 21 publications

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“…One approach is to use a nonuniform mesh (which must be appropriately chosen) which is very fine "in the boundary layer" and coarser elsewhere, e.g. [4], [7], [18], [22]. Another approach has been to devise schemes which have no formal cell Reynolds number limitation.…”

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confidence: 99%

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Generalized OCI Schemes for Boundary Layer Problems

Berger¹,

Solomon²,

Ciment³

et al. 1980

Mathematics of Computation

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“…Another approach has been to devise schemes which have no formal cell Reynolds number limitation. Schemes of this type have been constructed by using noncentered ("upwind") differencing for the first derivative term, or, more generally, by adding an "artificial viscosity" to the diffusion coefficient e, e.g., [1], [5], [7], [10], [11], [13], [28], [29].…”

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confidence: 99%

Generalized OCI Schemes for Boundary Layer Problems

Berger¹,

Solomon²,

Ciment³

et al. 1980

Mathematics of Computation

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“…The perturbed matrix A(δ) may or may not be singular for δ = 0. Some important applications where this problem arises are finite difference methods for the solution of singular perturbation problems [9,10], differential equations with singular constant coefficients [7], least square problems [26], asymptotic linear programming [21,22], and Markov chains [19,1,8].…”

Section: Introductionmentioning

confidence: 99%

Laurent expansion of the inverse of perturbed, singular matrices

González-Rodríguez¹,

Moscoso²,

Kindelán³

2015

Journal of Computational Physics

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a b s t r a c tKeywords: Laurent series, Inverse, Radial basis functions, Interpolation.In this paper we describe a numerical algorithm to compute the Laurent expansion of the inverse of singularly perturbed matrices. The algorithm is based on the resolvent formalism used in complex analysis to study the spectrum of matrices. The input of the algorithm are the matrix coefficients of the power series expansion of the perturbed matrix. The matrix coefficients of the Laurent expansion of the inverse are computed using recursive analytical formulae. We show that the computational complexity of the proposed algorithm grows algebraically with the size of the matrix, but exponentially with the order of the singularity. We apply this algorithm to several matrices that arise in applications. We make special emphasis to interpolation problems with radial basis functions.

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“…This difference operator gives an 0(h2) approximation to the differential equation, but is known to give poor results for small e [1]. The three figures give results of computations using h = 0.02 and for three different values of e. In Figures 1 and 2 we have plotted the errors in the approximate solutions.…”

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confidence: 98%

“…A useful discussion of a variety of problems is contained in Dorr [1 ]. II'in [2] gives an 0(h) error bound for his scheme that is uniform in e. We give a different proof of this result.…”

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confidence: 99%