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Recent Advances in the Study of a Fourth-Order Compact Scheme for the One-Dimensional Biharmonic Equation

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Abstract

It is well-known that non-periodic boundary conditions reduce considerably the overall accuracy of an approximating scheme. In previous papers the present authors have studied a fourth-order compact scheme for the one-dimensional biharmonic equation. It relies on Hermitian interpolation, using functional values and Hermitian derivatives on a three-point stencil. However, the fourth-order accuracy is reduced to a mere first-order near the boundary. In turn this leads to an “almost third-order” accuracy of the approximate solution. By a careful inspection of the matrix elements of the discrete operator, it is shown that the boundary does not affect the approximation, and a full (“optimal”) fourth-order convergence is attained. A number of numerical examples corroborate this effect.

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Acknowledgements

We would like to thank Professor B. Bialecki of the Colorado School of Mines, who challenged us with providing a proof for the fourth-order accuracy of the three point biharmonic operator.

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Authors and Affiliations

  1. Afeka - Tel-Aviv Academic College of Engineering, 218 Bnei-Efraim St., Tel-Aviv, 69107, Israel

    D. Fishelov

  2. Institute of Mathematics, The Hebrew University, Jerusalem, 91904, Israel

    M. Ben-Artzi

  3. Department of Mathematics, LMAM, UMR 7122, University of Paul Verlaine-Metz, Metz, 57045, France

    J.-P. Croisille

Authors
  1. D. Fishelov
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  2. M. Ben-Artzi
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  3. J.-P. Croisille
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Corresponding author

Correspondence to D. Fishelov.

Additional information

This paper is dedicated to Professor Saul Abarbanel on the occasion of his 80-th birthday.

The authors were partially supported by a French-Israeli scientific cooperation grant 3-1355. The first author was also supported by a research grant of Afeka - Tel-Aviv Academic College of Engineering.

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Fishelov, D., Ben-Artzi, M. & Croisille, JP. Recent Advances in the Study of a Fourth-Order Compact Scheme for the One-Dimensional Biharmonic Equation. J Sci Comput 53, 55–79 (2012). https://doi.org/10.1007/s10915-012-9611-x

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  • DOI: https://doi.org/10.1007/s10915-012-9611-x

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