An Optimal Finite Difference Scheme with Minimized Dispersion and Adaptive Dissipation Considering the Spectral Properties of the Fully Discrete Scheme

For the simulation of flow fields with a broad range of length scales, designing numerical schemes with good spectral properties is one of the most important issues. To improve the spectral properties of the semi-discrete finite different schemes, the authors have previously proposed the idea of optimizing the dispersion and dissipation properties separately and a class of finite difference scheme with minimized dispersion and controllable dissipation properties is thus developed (Sun et al. J Comput Phys 230:4616–4635, 2011, 270:238–254, 2014). In the present paper, we further investigated this idea and extend it to the fully discrete scheme. In other words, the dispersion and dissipation errors induced by the temporal discretization are taken into consideration in the present paper. Moreover, a scale sensor is designed in the control of dissipation error to ensure the numerical scheme can automatically adjust its dissipation according to the local characteristic of the flow field. To achieve the shock-capturing capability, this optimized scheme is blended with the WENO-Z scheme to form a hybrid scheme. A number of benchmark test cases including the transportation of a linear wave, the propagation of a sound wave packet, the Shu–Osher problem, the double Mach reflection problem and the Rayleigh–Taylor instability problem are employed to verify the good spectral properties and robust shock-capturing capability of the proposed scheme.