Abstract
We present high-order compact schemes for fourth-order time-dependent problems, which are related to the “buckling plate” or the “clamping plate” problems. Given a mesh size h, we show that the truncation error is O(h 4) at interior points and O(h) at near-boundary points. In addition, the convergence of these schemes is analyzed. Although the truncation error is only of first-order at near-boundary points, we have proved that the error of these schemes converges to zero as h tends to zero at least as O(h 3. 5). Numerical results are performed and they calibrate the high-order accuracy of the schemes. It is shown that the numerical rate of convergence is actually four, thus the error tends to zero as O(h 4).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
S. Abarbanel, A. Ditkowski, B. Gustafsson, On error bound of finite difference approximations for partial differential equations. J. Sci. Comput. 15, 79–116 (2000)
M. Ben-Artzi, J-P. Croisille, D. Fishelov, Convergence of a compact scheme for the pure streamfunction formulation of the unsteady Navier–Stokes system. SIAM J. Numer. Anal. 44, 1997–2024 (2006)
M. Ben-Artzi, J.-P. Croisille, D. Fishelov, A fast direct solver for the biharmonic problem in a rectangular grid. SIAM J. Sci. Comput. 31(1), 303–333 (2008)
M. Ben-Artzi, J.-P. Croisille, D. Fishelov, Navier-Stokes Equations in Planar Domains (Imperial College Press, London, 2013)
D. Fishelov, M. Ben-Artzi, J-P. Croisille, Recent advances in the study of a fourth-order compact scheme for the one-dimensional biharmonic equation. J. Sci. Comput. 53, 55–70 (2012)
B. Gustafsson, The convergence rate for difference approximations to mixed initial boundary value problems. Math. Comput. 29, 386–406 (1975)
B. Gustafsson, The convergence rate for difference approximations to general mixed initial boundary value problems. SIAM J. Numer. Anal. 18, 179–190 (1981)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Fishelov, D. (2015). Semi-discrete Time-Dependent Fourth-Order Problems on an Interval: Error Estimate. In: Abdulle, A., Deparis, S., Kressner, D., Nobile, F., Picasso, M. (eds) Numerical Mathematics and Advanced Applications - ENUMATH 2013. Lecture Notes in Computational Science and Engineering, vol 103. Springer, Cham. https://doi.org/10.1007/978-3-319-10705-9_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-10705-9_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-10704-2
Online ISBN: 978-3-319-10705-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)