Top

Journal of Scientific Computing

Published in:

01-10-2012

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

Authors: D. Fishelov, M. Ben-Artzi, J.-P. Croisille

Published in: Journal of Scientific Computing | Issue 1/2012

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

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.

previous article New Analysis of the Du Fort–Frankel Methods

next article High Order Weighted Essentially Non-oscillation Schemes for Two-Dimensional Detonation Wave Simulations

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Abarbanel, S., Ditkowski, A., Gustafsson, B.: On error bound of finite difference approximations for partial differential equations. J. Sci. Comput. 15(1), 79–116 (2000) MathSciNetMATHCrossRef

Altas, I., Dym, J., Gupta, M.M., Manohar, R.P.: Multigrid solution of automatically generated high-order discretizations for the biharmonic equation. SIAM J. Sci. Comput. 19, 1575–1585 (1998) MathSciNetMATHCrossRef

Ben-Artzi, M., Croisille, J.-P., Fishelov, D.: Convergence of a compact scheme for the pure streamfunction formulation of the unsteady Navier-Stokes system. SIAM J. Numer. Anal. 44(5), 1997–2024 (2006) MathSciNetMATHCrossRef

Ben-Artzi, M., Croisille, J.-P., Fishelov, D.: A fast direct solver for the biharmonic problem in a rectangular grid. SIAM J. Sci. Comput. 31(1), 303–333 (2008) MathSciNetMATHCrossRef

Brüger, A., Gustafsson, B., Lötstedt, P., Nilsson, J.: High order accurate solution of the incompressible Navier-Stokes equations. J. Comput. Phys. 203, 49–71 (2005) MathSciNetMATHCrossRef

Carey, G.F., Spotz, W.F.: High-order compact scheme for the stream-function vorticity equations. Int. J. Numer. Methods Eng. 38, 3497–3512 (1995) MathSciNetMATHCrossRef

Carey, G.F., Spotz, W.F.: Extension of high-order compact schemes to time dependent problems. Numer. Methods Partial Differ. Equ. 17(6), 657–672 (2001) MathSciNetMATHCrossRef

Carpenter, M.H., Gottlieb, D., Abarbanel, S.: The stability of numerical boundary treatments for compact high-order schemes finite difference schemes. J. Comput. Phys. 108, 272–295 (1993) MathSciNetMATHCrossRef

Weinan, E., Liu, J.-G.: Essentially compact schemes for unsteady viscous incompressible flows. J. Comput. Phys. 126, 122–138 (1996) MathSciNetMATHCrossRef

10.

Fishelov, D., Ben-Artzi, M., Croisille, J.-P.: Recent developments in the pure streamfunction formulation of the Navier-Stokes system. J. Sci. Comput. 45(1–3), 238–258 (2010) MathSciNetMATHCrossRef

11.

Gupta, M.M., Manohar, R.P., Stephenson, J.W.: Single cell high order scheme for the convection-diffusion equation with variable coefficients. Int. J. Numer. Methods Fluids 4, 641–651 (1984) MathSciNetMATHCrossRef

12.

Gustafsson, B.: The convergence rate for difference approximations to mixed initial boundary value problems. Math. Comput. 29, 396–406 (1975) MathSciNetMATHCrossRef

13.

Gustafsson, B.: The convergence rate for difference approximations to general mixed initial boundary value problems. SIAM J. Numer. Anal. 18, 179–190 (1981) MathSciNetMATHCrossRef

14.

Lele, S.K.: Compact finite-difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16–42 (1992) MathSciNetMATHCrossRef

15.

Li, Ming, Tang, Tao: A compact fourth-order finite difference scheme for unsteady viscous incompressible flows. J. Sci. Comput. 16(1), 29–45 (2001) MathSciNetMATHCrossRef

16.

Stephenson, J.W.: Single cell discretizations of order two and four for biharmonic problems. J. Comput. Phys. 55, 65–80 (1984) MathSciNetMATHCrossRef

Title: Recent Advances in the Study of a Fourth-Order Compact Scheme for the One-Dimensional Biharmonic Equation
Authors: D. Fishelov
M. Ben-Artzi
J.-P. Croisille
Publication date: 01-10-2012
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2012
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-012-9611-x

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 1/2012

Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering for Discontinuous Galerkin Solutions: Improved Errors Versus Higher-Order Accuracy

The Narrow Escape Problem—A Short Review of Recent Results

The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes

WENO Scheme with Subcell Resolution for Computing Nonconservative Euler Equations with Applications to One-Dimensional Compressible Two-Medium Flows

New Analysis of the Du Fort–Frankel Methods

Stability and Convergence Analysis of Fully Discrete Fourier Collocation Spectral Method for 3-D Viscous Burgers’ Equation

Premium Partner