A COMPARISON OF HIGHER-ORDER COMPACT FINITE-DIFFERENCE SCHEMES
Keywords:
High-Order Finite Difference, Burgers EquationAbstract
In this work, three fourth-order-accurate, compact finite-difference schemes, namely, the
Hermitian (H), the cell-centered Hermitian (CCH) and the Spotz-Carey (S-C) schemes are
studied. The schemes are described and their accuracies are investigated using the one dimensional viscous Burgers equation as the testing model. Comparisons for the absolute, the
average and the maximum relative errors are shown. The effect of spatial step size, h, on the
accuracy of the selected schemes is investigated. A new procedure, for using the CCH scheme is
proposed and found to produce the least error. The new procedure utilizes a combination of a
fift h- and a sixth-order interpolation schemes. Other properties of the schemes, such as
additional relations required and ease ofimplementation are also discussed.
References
Hirsh, R, 1975, "Higher Order Accurate Difference Solutions of Fluid Mechanics ProblemOs by a Compact Differencing Technique", Journal of Computational Physics, 19: 90-109 .
Adam, Y., 1977, "Highly Accurate Compact Implicit Methods and Boundary Conditions", Journal of Computational Physics; 24: 10-22.
Rubin S.G, and Khosla, PK, 1977, "Polynomial Interpolation Methods for Viscous Flow Calculations", Journal of Computational Physics, 24: 217-244.
Goedheer, W.J. and Potters, J.H.H.M. 1985, "A Compact Finite Difference Scheme on a Non-Equidistant Mesh", Journal of Computational Physics; 61: 269-279.
Lele, S.K, 1992, "Compact Finite Difference Schemes with Spectral-Like Resolution", Journal of Computational Physics ; 103: 16-42.
Cockburn, B. and Shu, C.W., 1994, "Nonlinearly Stable Compact Schemes for Shock Calculations", SIAM Journal on Numerical Analysis, 31: 607627.
Haras, Z. and Ta'asan, S., 1994, "Finite Difference Schemes for Long-Time Integration", Journal of Computational Physics, 114: 265-279.
Deng, X. and Maekawa, H., 1997, "Compact High-Order Accurate Nonlinear Schemes", Journal of Computational Physics 1997; 130: 77-91.
Ravichandran, K, 1997, "Higher Order KFVS Algorithms Using Compact Upwind Difference Operators", Journal of Computational Physics; 130: 161173.
Fu, D. and Ma, Y, 1997, "A High Order Accurate Difference Scheme for Complex flow Fields", Journal of Computational Physics; 134: 1-15.
Visbal, M.R and Gaitonde, D.V., 1999, "High-Order-Accurate Methods for Complex Unsteady Subsonic Flows", AIAA Journal; 10: 1231-1239.
Wilson, RV., Demuren, A.D. and Carpenter, M., 1998, "High-Order Compact Schemes for Numerical Simulation of Incompressible Flows", ICASE Report No. 98-13, NASNCR-1998-206922.
Ekaterinaris, J.A., 2000, "Implicit High-Order-Accurate-in-Space Algorithms for the Navier-Stokes Equations", AIAA Journal, 38: 1594-1602.
Reuter, J. and Rempfer, D., 2000, "High Order Vorticity-Velocity Methodfor the Simulation of Pipe Flow Transition", Applied Numerical Mathematics, 33: 105-111.
Spotz, W.F. and Carey, G.F., 1996, "High-Order Compact Finite Difference Methods", In Proceedings of the 2nd International Conference on Spectral and High Order Methods, University of Houston, PP. 397-407.
Spotz, W.F., 1995, "High-Order Compact Finite Difference Schemes for Computational Mechanics",. Ph. D. Dissertation, The University of Texas at Austin.
MacKinnon, RJ. and Carey, G.F., 1988, "Analysis of Material Interface Discontinuities and Superconvergent Fluxes in Finite Difference Theory", Journal of Computational Physics, 75: 151-167.
Abarbanel, S. and Kumar, A., 1988, "Compact High-Order Schemes for the Euler Equations", Journal of Scientific Computing, 3: 275-288.
Dennis, S.C.R and Hudson, J.D., 1989, "Compact h4 Finite-Difference Approximations to Operators of Navier-Stokes Type", Journal of Computational Physics, 85: 390-416.
Gupta, M.M., Manohar, RP. and Stephenson, JW., 1984, "A Single Cell High Order Scheme for the Convection-Diffusion Equation with Variable Coefficients", International Journal for Numerical Methods in Fluids, 4: 641651.
Asrar, W., Basri, S. and Arora P.R, 2000, "High Order Compact Solution of the Transient Diffusion Equation", Journal - Institution of Engineers, Malaysia, 61: 41-46.
Hoffmann, K.A. and Chiang, S.T., 1993, "Computational Fluid Dynamics for Engineers", I. Engineering Education System, USA, P 265.
Downloads
Published
How to Cite
Issue
Section
License
Copyright of articles that appear in Jurnal Mekanikal belongs exclusively to Penerbit Universiti Teknologi Malaysia (Penerbit UTM Press). This copyright covers the rights to reproduce the article, including reprints, electronic reproductions or any other reproductions of similar nature.
