A family of spatio-temporal optimized finite difference schemes with adaptive dispersion and critical-adaptive dissipation for compressible flows

Abstract

A numerical scheme with good dissipation and dispersion properties is essential for simulations of complex compressible flows in resolving small-scale structures and capturing discontinuities. Although a sufficient numerical dissipation benefits the shock-wave capturing, the fine scales in smooth regions are dissipated either. In this work, a framework to construct arbitrarily high-order spatio-temporal optimized finite difference schemes with adaptive dispersion and critical-adaptive dissipation is proposed, where the dispersion and dissipation properties are characterized by two free parameters respectively. The proposed scheme can automatically adjust these two free parameters to control the dispersion and dissipation according to the local flow-field properties quantified by the scale sensor. As the first step to optimize the spectral properties of the fully discrete scheme, the total dispersion and dissipation errors induced by spatio-temporal discretization are studied. Then, the dispersion property is optimized by a new integrated error function devised for fully discrete scheme. To improve the precision of prediction, the scale sensor employed to quantify the local scaled wavenumber of flow fields is optimized in wavenumber space. Furthermore, a modified dispersion-dissipation condition is developed to characterize the relationship between the total dispersion and dissipation errors. Building upon the optimized scale sensor and modified dispersion-dissipation condition, the critical-adaptive dissipation parameter is obtained, which keeps the numerical dissipation as low as possible to resolve more small-scale structures in the low-wavenumber region and introduces sufficient dissipation to capture strong discontinuities successfully. Moreover, the adaptive dispersion parameter related to the local wavenumber and Courant number is obtained to reduce the total dispersion error significantly. Meanwhile, the critical-adaptive dissipation surface and adaptive dispersion surface are constructed for different Courant number utilized in actual flow simulations, which improves the spectral properties of the proposed scheme. Finally, some benchmark cases involving strong discontinuities and multi-scale structures are employed to verify the attractive performance of the proposed scheme.