A class of high-order finite difference schemes with minimized dispersion and adaptive dissipation for solving compressible flows

For the compressible flows with broadband length scales, good dispersion and dissipation properties are crucial for the numerical schemes to realize high-fidelity simulations. It has been recognized that the minimization of dispersion error is an effective measure to improve the resolution of schemes. However, it is still an open problem for the schemes to control or adjust the dissipation. In this paper, a class of high-order finite difference schemes with minimized dispersion and adaptive dissipation is proposed. As the first step to automatically adjust the dissipation according to the flow structures, we devise a scale sensor to quantify the local length scale of the numerical solution as the effective scaled wavenumber. Then the dispersion-dissipation condition is used to construct the relationship between the dissipation parameter and the effective scaled wavenumber. Thus, we obtain a class of finite difference schemes with adaptive dissipation. To achieve the shock-capturing capability and maintain the superior spectral properties in the smooth regions, we combine the adaptive dissipation scheme with the corresponding weighted essentially non-oscillatory (WENO) scheme to construct a hybrid scheme. At each grid point, either of the sub-schemes is applied dynamically according to a new shock detector. The shock detection accuracy of the new detector has been effectively improved compared with classical ones. The approximate dispersion relation (ADR) shows that the new hybrid scheme is accurate and robust. Several benchmark test problems with broadband length scales as well as discontinuities are presented to verify the high resolution, high efficiency and the good shock-capturing capability of the proposed scheme. The performance of the proposed scheme in turbulence simulation is also demonstrated by the three-dimensional isotropic Taylor-Green vortex problem.