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A parameter-uniform numerical method for a Sobolev problem with initial layer

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Abstract

The present study is concerned with the numerical solution, using finite difference method of a one-dimensional initial-boundary value problem for a linear Sobolev or pseudo-parabolic equation with initial jump. In order to obtain an efficient method, to provide good approximations with independence of the perturbation parameter, we have developed a numerical method which combines a finite difference spatial discretization on uniform mesh and the implicit rule on Shishkin mesh(S-mesh) for the time variable. The fully discrete scheme is shown to be convergent of order two in space and of order one expect for a logarithmic factor in time, uniformly in the singular perturbation parameter. Some numerical results confirming the expected behavior of the method are shown.

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Authors and Affiliations

  1. Faculty of Art and Science, Department of Mathematics, Yüzüncü Yıl University, 65080, Van, Turkey

    G. M. Amiraliyev & Hakki Duru

  2. Faculty of Agriculture, Yüzüncü Yıl University, 65080, Van, Turkey

    I. G. Amiraliyeva

Authors
  1. G. M. Amiraliyev
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  2. Hakki Duru
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  3. I. G. Amiraliyeva
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Correspondence to G. M. Amiraliyev.

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Amiraliyev, G.M., Duru, H. & Amiraliyeva, I.G. A parameter-uniform numerical method for a Sobolev problem with initial layer. Numer Algor 44, 185–203 (2007). https://doi.org/10.1007/s11075-007-9096-0

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  • DOI: https://doi.org/10.1007/s11075-007-9096-0

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