The dispersion optimization and dissipation adjustment for high-order finite difference schemes

In this paper, our recent work on the dispersion optimization and dissipation adjustment for high-order finite difference schemes is reviewed. For compressible flows with a broad range of length scales, good dispersion and dissipation properties are crucial for 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, while the dissipation should be adjusted according to the local scale of the solution. We found a sufficient condition for a semi-discretized finite difference scheme to have independent dispersion and dissipation properties. Based on this finding, the high-order finite difference scheme with minimized dispersion and controllable dissipation (MDCD) was proposed. The MDCD scheme has been widely used and achieved good performance. However, one drawback is that the dissipation is controlled empirically. To realize the automatic adjustment of the dissipation according to the local scale of the solution, we devised a scale sensor to quantify the length scale of the numerical solution which can be calibrated to give the effective scaled wavenumbers. Then the dispersion-dissipation condition is used to construct the relationship between the dissipation parameter and the effective scaled wavenumber. A class of finite difference schemes with adaptive dissipation is thus obtained. To achieve the shock-capturing capability while maintaining the superior spectral properties in the smooth regions, we make an improvement to a classical shock detector by incorporating the proposed scale sensor, and construct a new hybrid scheme which consists of the adaptive dissipation scheme and the corresponding weighted essentially non-oscillation (WENO) scheme. 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 validate the high-resolution and the good shock-capturing capability of the proposed scheme.