An Overview of Some High Order and Multi-Level Finite Difference Schemes in Computational Aeroacoustics
In this paper, we have combined some spatial derivatives with the optimised time derivative proposed by Tam and Webb in order to approximate the linear advection equation which is given by = 0. Ôêé Ôêé + Ôêé Ôêé x f t u These spatial derivatives are as follows: a standard 7-point 6 th -order central difference scheme (ST7), a standard 9-point 8 th -order central difference scheme (ST9) and optimised schemes designed by Tam and Webb, Lockard et al., Zingg et al., Zhuang and Chen, Bogey and Bailly. Thus, these seven different spatial derivatives have been coupled with the optimised time derivative to obtain seven different finite-difference schemes to approximate the linear advection equation. We have analysed the variation of the modified wavenumber and group velocity, both with respect to the exact wavenumber for each spatial derivative. The problems considered are the 1-D propagation of a Boxcar function, propagation of an initial disturbance consisting of a sine and Gaussian function and the propagation of a Gaussian profile. It is known that the choice of the cfl number affects the quality of results in terms of dissipation and dispersion characteristics. Based on the numerical experiments solved and numerical methods used to approximate the linear advection equation, it is observed in this work, that the quality of results is dependent on the choice of the cfl number, even for optimised numerical methods. The errors from the numerical results have been quantified into dispersion and dissipation using a technique devised by Takacs. Also, the quantity, Exponential Error for Low Dispersion and Low Dissipation, eeldld has been computed from the numerical results. Moreover, based on this work, it has been found that when the quantity, eeldld can be used as a measure of the total error. In particular, the total error is a minimum when the eeldld is a minimum.
[1] A.R. Appadu, M.Z. Dauhoo and S.D.D.V. Rughooputh. Optimisation of
Numerical Schemes Using the Minimised Integrated Square Difference
Error. Research and Innovation Challenges. University of Mauritius, 16-
20 January 2007.
[2] A.R. Appadu, M.Z. Dauhoo and S.D.D.V. Rughooputh. Control of
Numerical Effects of Dispersion and Dissipation in Numerical Schemes
for Efficient Shock-Capturing Through an Optimal Courant Number.
Computers and Fluids, Vol. 37, No. 6, 2008, pp. 767-783.
[3] A.R. Appadu and M.Z. Dauhoo. A Note on the Two Versions of Lax-
Friedrichs Scheme. Proceedings of The 2007 International Conference
on Scientific Computing. CSREA Press, Editors: H. R. Arabnia and J. Y.
Yang and M. Q. Yang.
[4] A.R. Appadu, M.Z. Dauhoo and S.D.D.V. Rughooputh. Efficient Shock-
Capturing Numerical Schemes Using the Approach of Minimised
Integrated Square Difference Error for Hyperbolic Conservation Laws.
Proceedings of Computational Science and Its Applications- ICCSA
2007. Lecture Notes in Computer Science. Editors: O. Gervasi Osvaldo
and M.L. Gavrilova, Vol. III.
[5] A.R. Appadu and M.Z. Dauhoo. On the Concept of Minimised
Integrated Square Difference Error: Its Mechanism and Some of Its
Applications. Submitted to SIAM Journal of Numerical Analysis
(September 2008).
[6] A.R. Appadu and M.Z. Dauhoo. The Concept of Minimised Integrated
Exponential Error for Low Dispersion and Low Dissipation. Submitted
to International Journal for Numerical Methods in Fluids (September
2008).
[7] G. Ashcroft and X. Zhang. Optimised prefactored compact schemes,
Journal of Computational Physics, vol. 190, 2003, pp. 459-477.
[8] C. Bogey and C. Bailly. A Family of Low Dispersive and Low
Dissipative Explicit Schemes for Computing the Aerodynamic Noise.
AIAA-Paper 2002.
[9] C. Bogey and C. Bailly. A Family of Low Dispersive and Low
Dissipative Explicit Schemes for Flow and Noise Computations. Journal
of Computational Physics, 2004, vol. 194, pp. 194-214.
[10] Lodovica Ferraria.
http://www.sosmath.com/algebra/factor/fac12/fac12.html.
[11] R. Hixon. Evaluation of high-accuracy MacCormack-Type scheme using
Benchmark Problems, NASA Contractor Report 202324, ICOMP-97-03-
1997.
[12] F.Q. Hu, M.Y. Hussaini and J. Manthey. Low-Dissipation and
Dispersion Runge-Kutta Schemes For Computational Acoustics.
Technical Report: TR-94-102, 1994.
[13] S. Johansson. High Order Finite Difference Operators with the
Summation by Parts Property based on DRP Schemes, Division of
Scientific Computing-Department of Information Technology, Uppsala
University, Sweden, 2007.
[14] D.P. Lockard, K.S. Brentner and H.L. Atkins. High-accuracy algorithms
for computational aeroacoustics. AIAA Journal, vol. 33, No. 2, pp. 246-
251, 1995.
[15] M. Popescu and W. Shyy. Dispersion-Relation-Preserving and Space-
Time schemes for Wave Equations, AIAA, Paper No. 202-0225, 2002.
[16] P. Roe. Linear Bicharacteristics schemes without dissipation. SIAM
Journal of Scientific Computing, vol. 19, No. 5, (1998), pp. 1405-1429.
[17] W. De Roeck, M. Baelmans, and P. Sas. An overview of high-order
finite difference schemes for computational aeroacoustics, International
Conference on Noise and Vibration Engineering. Katholieke Universiteit
Leuven, Belgium, ISMA 20-22 September 2004, pp. 353-368.
[18] T.K. Sengupta, G. Ganeriwal and S. De. Analysis of Central and Upwind
Compact Schemes. Journal of Computational Physics, Vol. 192, 2003,
pp. 677-694.
[19] C.K.W. Tam and J.C. Webb. Dispersion-Relation-Preserving Finite
Difference Schemes for Computational Acoustics. Journal of
Computational Physics, Vol. 107, 1993, pp. 262-281.
[20] C.K.W. Tam. Computational Aeroacoustics: Issues and Methods.
AIAA, 33, 10, pp. 1788-1796, October 1995.
[21] L. Takacs. A Two-step scheme for the Advection Equation with
Minimized Dissipation and Dispersion errors. Monthly Weather
Review. 113, (1985), pp. 1050-1065.
[22] D. J. Webb, B.A. De Cuevas and C.S. Richmond. Improved Advection
Schemes for Ocean Models. Journal of Atmospheric and Oceanic
Technology. Vol. 15, No. 5, 1998, pp. 1171-1187.
[23] V.L.Wells and R.A.Renault. Computing Aerodynamically Generated
Noise, Annual Rev. Fluid Mechanics, vol. 29, 1997, pp.161-199.
[24] D.W. Zingg, H. Lomax and H.M. Jurgens. High-Accuracy finite
difference schemes for linear wave propagation. SIAM Journal of
Scientific Computing, vol. 17, no. 2, 1996, pp. 328-346.
[25] D.W. Zingg. Comparison of High-Accuracy Finite-Difference Methods
for Linear Wave Propagation. SIAM Journal of Scientific Computing.
Vol. 22,no. 2, pp. 476-502, 2001.
[26] M. Zhuang and R.F. Chen. Application of high-order optimised upwind
schemes for computational aeroacoustics. AIAA Journal, vol. 40,No. 3,
2002, pp. 443-449.
[1] A.R. Appadu, M.Z. Dauhoo and S.D.D.V. Rughooputh. Optimisation of
Numerical Schemes Using the Minimised Integrated Square Difference
Error. Research and Innovation Challenges. University of Mauritius, 16-
20 January 2007.
[2] A.R. Appadu, M.Z. Dauhoo and S.D.D.V. Rughooputh. Control of
Numerical Effects of Dispersion and Dissipation in Numerical Schemes
for Efficient Shock-Capturing Through an Optimal Courant Number.
Computers and Fluids, Vol. 37, No. 6, 2008, pp. 767-783.
[3] A.R. Appadu and M.Z. Dauhoo. A Note on the Two Versions of Lax-
Friedrichs Scheme. Proceedings of The 2007 International Conference
on Scientific Computing. CSREA Press, Editors: H. R. Arabnia and J. Y.
Yang and M. Q. Yang.
[4] A.R. Appadu, M.Z. Dauhoo and S.D.D.V. Rughooputh. Efficient Shock-
Capturing Numerical Schemes Using the Approach of Minimised
Integrated Square Difference Error for Hyperbolic Conservation Laws.
Proceedings of Computational Science and Its Applications- ICCSA
2007. Lecture Notes in Computer Science. Editors: O. Gervasi Osvaldo
and M.L. Gavrilova, Vol. III.
[5] A.R. Appadu and M.Z. Dauhoo. On the Concept of Minimised
Integrated Square Difference Error: Its Mechanism and Some of Its
Applications. Submitted to SIAM Journal of Numerical Analysis
(September 2008).
[6] A.R. Appadu and M.Z. Dauhoo. The Concept of Minimised Integrated
Exponential Error for Low Dispersion and Low Dissipation. Submitted
to International Journal for Numerical Methods in Fluids (September
2008).
[7] G. Ashcroft and X. Zhang. Optimised prefactored compact schemes,
Journal of Computational Physics, vol. 190, 2003, pp. 459-477.
[8] C. Bogey and C. Bailly. A Family of Low Dispersive and Low
Dissipative Explicit Schemes for Computing the Aerodynamic Noise.
AIAA-Paper 2002.
[9] C. Bogey and C. Bailly. A Family of Low Dispersive and Low
Dissipative Explicit Schemes for Flow and Noise Computations. Journal
of Computational Physics, 2004, vol. 194, pp. 194-214.
[10] Lodovica Ferraria.
http://www.sosmath.com/algebra/factor/fac12/fac12.html.
[11] R. Hixon. Evaluation of high-accuracy MacCormack-Type scheme using
Benchmark Problems, NASA Contractor Report 202324, ICOMP-97-03-
1997.
[12] F.Q. Hu, M.Y. Hussaini and J. Manthey. Low-Dissipation and
Dispersion Runge-Kutta Schemes For Computational Acoustics.
Technical Report: TR-94-102, 1994.
[13] S. Johansson. High Order Finite Difference Operators with the
Summation by Parts Property based on DRP Schemes, Division of
Scientific Computing-Department of Information Technology, Uppsala
University, Sweden, 2007.
[14] D.P. Lockard, K.S. Brentner and H.L. Atkins. High-accuracy algorithms
for computational aeroacoustics. AIAA Journal, vol. 33, No. 2, pp. 246-
251, 1995.
[15] M. Popescu and W. Shyy. Dispersion-Relation-Preserving and Space-
Time schemes for Wave Equations, AIAA, Paper No. 202-0225, 2002.
[16] P. Roe. Linear Bicharacteristics schemes without dissipation. SIAM
Journal of Scientific Computing, vol. 19, No. 5, (1998), pp. 1405-1429.
[17] W. De Roeck, M. Baelmans, and P. Sas. An overview of high-order
finite difference schemes for computational aeroacoustics, International
Conference on Noise and Vibration Engineering. Katholieke Universiteit
Leuven, Belgium, ISMA 20-22 September 2004, pp. 353-368.
[18] T.K. Sengupta, G. Ganeriwal and S. De. Analysis of Central and Upwind
Compact Schemes. Journal of Computational Physics, Vol. 192, 2003,
pp. 677-694.
[19] C.K.W. Tam and J.C. Webb. Dispersion-Relation-Preserving Finite
Difference Schemes for Computational Acoustics. Journal of
Computational Physics, Vol. 107, 1993, pp. 262-281.
[20] C.K.W. Tam. Computational Aeroacoustics: Issues and Methods.
AIAA, 33, 10, pp. 1788-1796, October 1995.
[21] L. Takacs. A Two-step scheme for the Advection Equation with
Minimized Dissipation and Dispersion errors. Monthly Weather
Review. 113, (1985), pp. 1050-1065.
[22] D. J. Webb, B.A. De Cuevas and C.S. Richmond. Improved Advection
Schemes for Ocean Models. Journal of Atmospheric and Oceanic
Technology. Vol. 15, No. 5, 1998, pp. 1171-1187.
[23] V.L.Wells and R.A.Renault. Computing Aerodynamically Generated
Noise, Annual Rev. Fluid Mechanics, vol. 29, 1997, pp.161-199.
[24] D.W. Zingg, H. Lomax and H.M. Jurgens. High-Accuracy finite
difference schemes for linear wave propagation. SIAM Journal of
Scientific Computing, vol. 17, no. 2, 1996, pp. 328-346.
[25] D.W. Zingg. Comparison of High-Accuracy Finite-Difference Methods
for Linear Wave Propagation. SIAM Journal of Scientific Computing.
Vol. 22,no. 2, pp. 476-502, 2001.
[26] M. Zhuang and R.F. Chen. Application of high-order optimised upwind
schemes for computational aeroacoustics. AIAA Journal, vol. 40,No. 3,
2002, pp. 443-449.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:52748", author = "Appanah Rao Appadu and Muhammad Zaid Dauhoo", title = "An Overview of Some High Order and Multi-Level Finite Difference Schemes in Computational Aeroacoustics", abstract = "In this paper, we have combined some spatial derivatives with the optimised time derivative proposed by Tam and Webb in order to approximate the linear advection equation which is given by = 0. Ôêé Ôêé + Ôêé Ôêé x f t u These spatial derivatives are as follows: a standard 7-point 6 th -order central difference scheme (ST7), a standard 9-point 8 th -order central difference scheme (ST9) and optimised schemes designed by Tam and Webb, Lockard et al., Zingg et al., Zhuang and Chen, Bogey and Bailly. Thus, these seven different spatial derivatives have been coupled with the optimised time derivative to obtain seven different finite-difference schemes to approximate the linear advection equation. We have analysed the variation of the modified wavenumber and group velocity, both with respect to the exact wavenumber for each spatial derivative. The problems considered are the 1-D propagation of a Boxcar function, propagation of an initial disturbance consisting of a sine and Gaussian function and the propagation of a Gaussian profile. It is known that the choice of the cfl number affects the quality of results in terms of dissipation and dispersion characteristics. Based on the numerical experiments solved and numerical methods used to approximate the linear advection equation, it is observed in this work, that the quality of results is dependent on the choice of the cfl number, even for optimised numerical methods. The errors from the numerical results have been quantified into dispersion and dissipation using a technique devised by Takacs. Also, the quantity, Exponential Error for Low Dispersion and Low Dissipation, eeldld has been computed from the numerical results. Moreover, based on this work, it has been found that when the quantity, eeldld can be used as a measure of the total error. In particular, the total error is a minimum when the eeldld is a minimum.
", keywords = "Optimised time derivative, dissipation, dispersion, cfl number, Nomenclature: k : time step, h : spatial step, β :advection velocity, r: cfl/Courant number, hkrβ= , w =θ, h : exact wave number, n :time level, RPE : Relative phase error per unit time step, AFM :modulus of amplification factor", volume = "3", number = "2", pages = "79-16", }
