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Uniformly convergent difference schemes for singularly perturbed parabolic diffusion-convection problems without turning points

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Summary

We consider first the initial-boundary value problem for the parabolic equation

$$\varepsilon y_{xx} + a(x,t)y_x - b(x,t)y - d(x,t)y_t = f(x,t)$$

on [0, 1]×[0,T], where 0<ɛ≦1 anda, b, d, f are smooth witha>0,b>0 andd>0. Bounds ony and its derivatives are obtained under certain compatibility assumptions on the data. We examine a family of difference schemes for this problem which is exponentially fitted in thex-variable and uses classical differencing in thet-variable, on rectangular grids which are arbitrarily spaced in both thex andt directions. Under a Courant-Friedrichs-Lewy-type condition, the errors at the grid points are shown to be bounded byC(H+K), whereH(K) is the maximum mesh width in thex(t) direction andC is a constant independent of ɛ,H andK. The corresponding result for a two-point boundary value problem is also derived.

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Authors and Affiliations

  1. Mathematics Department, University College, Cork, Ireland

    Martin Stynes

  2. Dundalk Regional Technical College, Dundalk, Ireland

    Eugene O'Riordan

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  1. Martin Stynes
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  2. Eugene O'Riordan
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Research partly supported by Arts Research Fund, University College, Cork

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Stynes, M., O'Riordan, E. Uniformly convergent difference schemes for singularly perturbed parabolic diffusion-convection problems without turning points. Numer. Math. 55, 521–544 (1989). https://doi.org/10.1007/BF01398914

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