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Analyses of the Dispersion Overshoot and Inverse Dissipation of the High-Order Finite Difference Scheme

Published online by Cambridge University Press:  03 June 2015

Qin Li ,
Qilong Guo  and
Hanxin Zhang
Qin Li*
Affiliation:
State Key Laboratory of Aerodynamics, Mianyang 621000, Sichuan, China National Laboratory of Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Qilong Guo
Affiliation:
State Key Laboratory of Aerodynamics, Mianyang 621000, Sichuan, China
Hanxin Zhang
Affiliation:
State Key Laboratory of Aerodynamics, Mianyang 621000, Sichuan, China National Laboratory of Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
*
*Corresponding author. Email: qin-li@vip.tom.com
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Abstract

Analyses were performed on the dispersion overshoot and inverse dissipation of the high-order finite difference scheme using Fourier and precision analysis. Schemes under discussion included the pointwise- and staggered-grid type, and were presented in weighted form using candidate schemes with third-order accuracy and three-point stencil. All of these were commonly used in the construction of difference schemes. Criteria for the dispersion overshoot were presented and their critical states were discussed. Two kinds of instabilities were studied due to inverse dissipation, especially those that occur at lower wave numbers. Criteria for the occurrence were presented and the relationship of the two instabilities was discussed. Comparisons were made between the analytical results and the dispersion/dissipation relations by Fourier transformation of typical schemes. As an example, an application of the criteria was given for the remedy of inverse dissipation in Weirs & Martín’s third-order scheme.

Keywords

76M2074P99High-order difference schemedispersion overshootinverse dissipation
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Issue 6 , December 2013 , pp. 809 - 824
Copyright
Copyright © Global-Science Press 2013

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References

[1]Vichnevetsky, R. and Bowles, J. B., Fourier Analysis of Numerical Approximations of Hyperbolic Equations, SIAM, Philadelphia, 1982.Google Scholar
[2]Lele, S. K., Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103 (1992), pp. 16–42.Google Scholar
[3]Tam, C. K. W. and Webb, J. C., Dispersion-relation-preserving finite difference schemes for computational acoustics, J. Comput. Phys., 107 (1993), pp. 262–281.CrossRefGoogle Scholar
[4]Lockard, D. P., Brentner, K. S. and Atkins, H. L., High-accuracy algorithms for computational aeroacoustics, AIAA J., 33 (1995), pp. 246–251.CrossRefGoogle Scholar
[5]Adams, N. A. and Shariff, K., A high-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems, J. Comput. Phys., 127 (1996), pp. 27–51.CrossRefGoogle Scholar
[6]Weirs, V. G. and Candler, G. V., Optimization of weighted ENO schemes for DNS of compressible turbulence, AIAA 97-1940, 1997.Google Scholar
[7]Martín, M.P., Taylor, E. M., Wu, M. and Weirs, V. G., A bandwidth-optimized weno scheme for the direct numerical simulation of compressible turbulence, J. Comput. Phys., 220 (2006), pp. 270–289.Google Scholar
[8]Shu, C. W. and Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comput. Phys., 77 (1988), pp. 439–471.Google Scholar
[9]Shu, C. W., Essentially non-oscillatory and weighted essentially, non-oscillatory schemes for hyperbolic, Conservation Laws, NASA/CR-97-206253, ICASE Report No. 97-65.Google Scholar
[10]Deng, X. G. and Maekawa, H., Compact high-order accurate nonlinear schemes, J. Comput. Phys., 130 (1997), pp. 77–91.Google Scholar
[11]Deng, X. G. and Zhang, H. X., Developing high-order weighted compact nonlinear schemes, J. Comput. Phys., 165 (2000), pp. 22–44.Google Scholar