A Class of Finite Difference Schemes with Adaptive Controllable Dispersion and Low Dissipation for Compressible Turbulence

The dispersion and dissipation properties of a numerical scheme are critical in simulating flow fields involving a wide range of length scales. In this study, we highlight the common oversight of focusing merely on controlling dispersion error without considering the importance of appropriate dispersion and scalability in computational efficiency. This study demonstrates that adjusting dispersion to match the local flow field near discontinuities is more effective in suppressing oscillations than simply minimizing dispersion. This proposed high-order finite difference scheme with adaptive dispersion minimized dissipation (ADMD) achieves adaptive controllable dispersion near flow field discontinuities, known as the ADMD scheme. This scheme, derived as a fourth-order finite difference scheme with seven points based on Taylor expansion, comprises a basic central component, additional dissipation component, and dispersion component. By exploring the effect of dispersion on numerical oscillations and the importance of adjusting dispersion according to the local flow field, a discontinuity detection function was established to enable the dispersion properties to adapt to the local flow field. Drawing inspiration from flow field smoothing in the weighted essentially non-oscillatory (WENO) scheme, efforts were made to minimize scheme dissipation. The main benefits of the ADMD scheme over several WENO-type schemes are robustness and efficiency, as the ADMD scheme saves at least 40–90% CPU time compared to the same-order WENO-type schemes for some numerical examples. Additionally, the numerical scheme proves advantageous in terms of simulating the decaying isotropic turbulence problem of three-dimensional compressible turbulence.