Continuous and Numerical Analysis of a Multiple Boundary Turning Point Problem

Vulanović, Relja; Farrell, Paul A.

doi:10.1137/0730073

Cited by 42 publications

(24 citation statements)

References 9 publications

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“…Note that we make no assertions about a steady-state solution of this problem. Hence, the work of [13] is not relevant to this paper. The corresponding reduced problem is defined to be…”

Section: Introductionmentioning

confidence: 94%

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A Fitted Mesh Method for a Class of Singularly Perturbed Parabolic Problems with a Boundary Turning Point

Dunne

O’Riordan

Шишкин

2003

Computational Methods in Applied Mathematics

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A class of singularly perturbed time-dependent convection-diffusion problems with a boundary turning point is examined on a rectangular domain. The solution of problems from this class possesses a parabolic boundary layer in the neighborhood of one of the sides of the domain. Classical numerical methods on uniform meshes are known to be inadequate for problems with boundary layers. A numerical method consisting of a standard upwind finite difference operator on a fitted mesh is constructed. It is proved that the numerical approximations generated by this method converge uniformly with respect to the singular perturbation parameter. Numerical results are presented that verify computationally the theoretical result.2000 Mathematics Subject Classification: 65M06, 65M12, 65M15.

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“…Note that we make no assertions about a steady-state solution of this problem. Hence, the work of [13] is not relevant to this paper. The corresponding reduced problem is defined to be…”

Section: Introductionmentioning

confidence: 94%

“…This fact motivated the present authors' study of the problem. Ordinary differential equations of a form related to (1.1) have been dealt with by several authors (see, for example, [5,6,12,13]) and arise in geophysics and in modeling thermal boundary layers in laminar flow (see [12] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

A Fitted Mesh Method for a Class of Singularly Perturbed Parabolic Problems with a Boundary Turning Point

Dunne

O’Riordan

Шишкин

2003

Computational Methods in Applied Mathematics

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“…The problem analysed in Lemma 1.12, where the coefficient of u has a simple zero, has a simple turning point at x = 0. For multiple turning-point problems, where the coefficient of u has a multiple zero, less is known; see [VF93], where such a problem is discussed. There are few stability estimates for turning-point problems in the literature; see [Doe98] for some (L ∞ , L ∞ ) and (L 1 , L 1 ) estimates in certain situations for simple turning points.…”

Section: Linear Second-order Turning-point Problemsmentioning

confidence: 99%

Robust Numerical Methods for Singularly Perturbed Differential Equations

2008

Springer Series in Computational Mathematics

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“…Shishkin [1] and Dunne et al [16] constructed parameter uniform finite difference schemes with the use of the method of condensing grids, for a class of singularly perturbed parabolic problem with a boundary turning point. Vulanović and Farrell [17,18] analyse these turning point problem related to the problem class (1.1) for ordinary differential equation. Recently, Kadalbajoo et al [10] construct a parameter uniform B-spline collocation method for a singularly perturbed parabolic problem without turning point.…”

Section: Introductionmentioning

confidence: 99%

A layer adaptive B-spline collocation method for singularly perturbed one-dimensional parabolic problem with a boundary turning point

Gupta

Kadalbajoo

2010

Numer. Methods Partial Differential Eq.

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In this article, we develop a parameter uniform numerical method for a class of singularly perturbed parabolic equations with a multiple boundary turning point on a rectangular domain. The coefficient of the first derivative with respect to x is given by the formula a 0 (x, t)x p , where a 0 (x, t) ≥ α > 0 and the parameter p ∈ [1, ∞) takes the arbitrary value. For small values of the parameter ε, the solution of this particular class of problem exhibits the parabolic boundary layer in a neighborhood of the boundary x = 0 of the domain. We use the implicit Euler method to discretize the temporal variable on uniform mesh and a B-spline collocation method defined on piecewise uniform Shishkin mesh to discretize the spatial variable. Asymptotic bounds for the derivatives of the solution are established by decomposing the solution into smooth and singular component. These bounds are applied in the convergence analysis of the proposed scheme on Shishkin mesh. The resulting method is boundary layer resolving and has been shown almost second-order accurate in space and first-order accurate in time. It is also shown that the proposed method is uniformly convergent with respect to the singular perturbation parameter ε. Some numerical results are given to confirm the predicted theory and comparison of numerical results made with a scheme consisting of a standard upwind finite difference operator on a piecewise uniform Shishkin mesh.

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Continuous and Numerical Analysis of a Multiple Boundary Turning Point Problem

Cited by 42 publications

References 9 publications

A Fitted Mesh Method for a Class of Singularly Perturbed Parabolic Problems with a Boundary Turning Point

A Fitted Mesh Method for a Class of Singularly Perturbed Parabolic Problems with a Boundary Turning Point

Robust Numerical Methods for Singularly Perturbed Differential Equations

A layer adaptive B-spline collocation method for singularly perturbed one-dimensional parabolic problem with a boundary turning point

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