The convergence of a fourth-order compact scheme to the one-dimensional biharmonic problem is established in the case of general Dirichlet boundary conditions. The compact scheme invokes value of the unknown function as well as Pade approximations of its first-order derivative. Using the Pade approximation allows us to approximate the first-order derivative within fourth-order accuracy. However, although the truncation error of the discrete biharmonic scheme is of fourth-order at interior point, the truncation error drops to first-order at near-boundary points. Nonetheless, we prove that the scheme retains its fourth-order (optimal) accuracy. This is done by a careful inspection of the matrix elements of the discrete biharmonic operator. A number of numerical examples corroborate this effect. We also present a study of the eigenvalue problem . We compute and display the eigenvalues and the eigenfunctions related to the continuous and the discrete problems. By the positivity of the eigenvalues, one can deduce the stability of of the related time-dependent problem . In addition, we study the eigenvalue problem . This is related to the stability of the linear time-dependent equation . Its continuous and discrete eigenvalues and eigenfunction (or eigenvectors) are computed and displayed graphically.
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26 September 2012
NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2012: International Conference of Numerical Analysis and Applied Mathematics
19–25 September 2012
Kos, Greece
Research Article|
September 26 2012
Fourth-order convergence of a compact scheme for the one-dimensional biharmonic equation
D. Fishelov;
D. Fishelov
Afeka - Tel-Aviv Academic College of Engineering, 218 Bnei-Efraim St., Tel-Aviv 69107,
Israel
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M. Ben-Artzi;
M. Ben-Artzi
The Hebrew University, Israel, Jerusalem 91904, Israela, Institute of Mathematics,
Israel
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J.-P. Croisille
J.-P. Croisille
Department of Mathematics, LMAM, UMR 7122, University of Paul Verlaine-Metz,
France
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AIP Conf. Proc. 1479, 1101–1104 (2012)
Citation
D. Fishelov, M. Ben-Artzi, J.-P. Croisille; Fourth-order convergence of a compact scheme for the one-dimensional biharmonic equation. AIP Conf. Proc. 26 September 2012; 1479 (1): 1101–1104. https://doi.org/10.1063/1.4756339
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