Appanah Rao Appadu and Muhammad Zaid Dauhoo
An Overview of Some High Order and MultiLevel Finite Difference Schemes in Computational Aeroacoustics
78 - 93
2009
3
2
International Journal of Physical and Mathematical Sciences
https://publications.waset.org/pdf/3635
https://publications.waset.org/vol/26
World Academy of Science, Engineering and Technology
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. &Ocirc;&ecirc;&eacute; &Ocirc;&ecirc;&eacute; &Ocirc;&ecirc;&eacute; &Ocirc;&ecirc;&eacute; x f t u These spatial derivatives are as follows a standard 7point 6 th order central difference scheme (ST7), a standard 9point 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 finitedifference 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 1D 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.
Open Science Index 26, 2009