Special Meshes for Finite Difference Approximations to an Advection-Diffusion Equation with Parabolic Layers

In this paper we solve numerically two singularly perturbed linear convection±dif-fusion problems for heat transfer in a¯uid with an assumed¯ow ®eld in the neighbourhood of a 180°bend in a channel. In the ®rst problem the theoretical solution has a parabolic boundary layer and in the second problem there is both a parabolic and a regular boundary layer in the solution. The numerical method uses piecewise uniform ®tted meshes condensing in a neighbourhood of each boundary layer and a standard upwind ®nite dierence operator satisfying a discrete maximum principle. The numerical results con®rm computationally that the method is e-uniform in the sense that the rate of convergence and the error constant of the method are independent of the singular perturbation parameter e, where e denotes the reciprocal of the P eclet number of the¯uid. This e-uniform behaviour is obtained only when an appropriate piecewise uniform ®tted mesh is constructed for each boundary layer. This is con®rmed by several additional computations on meshes which do not ful®ll this requirement. Ó 2001 Elsevier Science Inc. All rights reserved.

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“…[11]). A numerical method for solving (2.1a)±(2.1g) is said to have an e-uniform rate of convergence p on the sequence of meshes fX…”

Section: Numerical Results On Uniform Meshesmentioning

confidence: 99%

Numerical experiments for advection–diffusion problems in a channel with a 180° bend

Clavero

Miller

O’Riordan

et al. 2001

Applied Mathematics and Computation

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“…This has been shown for linear problems, e.g. in Reference [1] where it is also seen that inappropriate condensing of the mesh in the boundary layer region also fails to resolve the di culty.…”

Section: Introductionmentioning

confidence: 88%

“…A monotone di erence method is required, in particular for the jet problem, but we still encounter stability di culties and thus need to generalize the algorithm from [1], as elaborated below. After linearization and discretization of (2) and the associated boundary conditions (8) we have the sequence of discrete linear problems for m = 0; 1; : : ::…”

Section: Introductionmentioning

confidence: 99%

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Parameter‐uniform numerical methods for a laminar jet problem

Ansari

Hegarty

Шишкин

2003

Numerical Methods in Fluids

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SUMMARYWe consider the classical problem of a two-dimensional laminar jet of incompressible uid owing into a stationary medium of the same uid. The equations of motion are the same as the boundary layer equations for ow past an inÿnite at plate, but with di erent boundary conditions. Numerical experiments show that, using appropriate piecewise-uniform meshes, numerical solutions together with their scaled discrete derivatives are obtained which are parameter (i.e., viscosity ) robust with respect to both the number of mesh nodes and the number of iterations required for convergence. While the method employed is non-conservative, we show with the aid of numerical experiments that the loss in conservation of momentum is minimal.

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“…In the paper [24] of Ross and Stynes, uniform convergence of an upwind type method is proved. For two-dimensional stationary problems, the papers of Hegarty et al [8,9] and Clavero et al [4], present some numerical results obtained using two-dimensional Shishkin meshes for regular and parabolic layers. The book of Miller et al [16] gives the most recent results on numerical approximation of singularly perturbed problems on Shishkin meshes.…”

Section: Introductionmentioning

confidence: 99%

A fractional step method on a special mesh for the resolution of multidimensional evolutionary convection-diffusion problems

Clavero

Jorge

Lisbona

et al. 1998

Applied Numerical Mathematics

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In this paper we consider numerical schemes for multidimensional evolutionary convection-diffusion problems, where the approximation properties are uniform in the diffusion parameter. In order to obtain an efficient method, to provide good approximations with independence of the size of the diffusion parameter, we have developed a numerical method which combines a finite difference spatial discretization on a special mesh and a fractional step method for the time variable. The special mesh allows a correct approximation of the solution in the boundary layers, while the fractional steps permits a low computational cost algorithm. Some numerical examples confirming the expected behavior of the method are shown.

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Special Meshes for Finite Difference Approximations to an Advection-Diffusion Equation with Parabolic Layers

Cited by 38 publications

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Numerical experiments for advection–diffusion problems in a channel with a 180° bend

Numerical experiments for advection–diffusion problems in a channel with a 180° bend

Parameter‐uniform numerical methods for a laminar jet problem

A fractional step method on a special mesh for the resolution of multidimensional evolutionary convection-diffusion problems

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