Spectral properties optimization is an important issue for developing schemes to resolve flow fields that are characterized by a wide range of length scales such as turbulent flows and aero acoustic phenomena. The work finished by Dr SUN Zhensheng and his ground provided a novel approach to optimize the dissipation and dispersion properties of a family of tri-diagonal compact schemes. The corresponding paper entitled “A high-resolution, hybrid compact-WENO scheme with minimized dispersion and controllable dissipation” was published in Sci China-Phys Mech Astron, 2014 Vol. 57(5).

For the detailed and high-fidelity simulation of compressible flows with a broadband of length scales, the numerical scheme should have spectral-like properties to resolve these small scales. Spectral properties optimization is an effective way to construct schemes with good spectral properties. Concerning the optimization of the spectral properties of a scheme, it is generally accepted the dispersion error should be minimized due to some chosen criteria. However, there are no general guidelines on how the dissipation should be optimized. In the previous work for the optimization of the spectral properties of a scheme, the cost functions are the blending of the dissipation and dispersion errors. As a result, the change of the dissipation properties will deteriorate the already optimized dispersion. Therefore, it would be beneficial if we can control the dissipation and dispersion properties of a scheme separately.

To design such kinds of schemes, a sufficient condition for a family of tri-diagonal compact schemes to have independent dispersion and dissipation is firstly derived. Then, a specific a fourth order compact scheme with minimized dispersion and controllable dissipation (MDCD) properties is constructed and analyzed. The dispersion properties of the corresponding scheme are optimized by minimizing an elaborately designed integrated error function while the dissipation is controlled by variation of one free parameter. Finally, the optimized compact scheme is blended with the wide-used weighted essentially non-oscillatory (WENO) scheme to handle the discontinuities for the practical use. Moreover, the approximation dispersion relation (ADR) approach is employed to study the spectral properties of the hybrid scheme to yield the true wave propagation behavior of the nonlinear scheme. The dispersion and dissipation curves of the proposed scheme are shown in the following figure.

(a) dispersion curves                                             (b) dissipation curves

Comparisons of the dispersion and dissipation properties for different schemes. WENO5 means the original 5^th order WENO scheme while MDCD-HY means the explicit MDCD schemes. Differentshows the dispersion curves of the scheme for different optimized. Differentshows the change of the dissipation properties by variation of.

The main novelty of the present research is the idea of designing a family of tri-diagonal compact schemes with independent dissipation and dispersion properties. The most important feature of such kinds of scheme is the adjustment of the dissipation will not affect the already optimized dispersion properties. Several benchmark test cases which contains both a broadband of length scales and discontinuities have been carried out to validate the flexible, robustness and less time-consuming of the proposed scheme.

This research was partially supported by a grant from the National Natural Science Foundation of Chinaand a National University Research Grant from Xi’an Research Institute of High-tech. The idea of controlling dissipation and dispersion separately is a breakthrough for the designing of the spectral-like schemes. The researchers suggested this approach to be absorbed in the in-house code for the direct numerical simulation (DNS) and large eddy simulation (LES) of compressible turbulence. The proposed scheme will be effective in extending the well-resolved wave numbers range and more small scale turbulent structures can be captured.

See the article: Sun Z S, Hu Y, Luo L, et al. A high-resolution, hybrid compact-WENO scheme with minimized dispersion and controllable dissipation. Sci China-Phys Mech Astron, 2014 Vol. 57(5): 971-982.

http://phys.scichina.com:8083/sciGe/EN/abstract/abstract508376.shtml