A class of order-preserving, spectral-property-optimized, and shock-capturing finite difference schemes for compressible flows

The numerical simulation of compressible flows, characterized by a wide range of flow length scales and discontinuities, still pose an open challenge to design the optimal schemes which resolve the small-scale flow structures with high-fidelity and capture the shock waves with no oscillations. In this paper, we proposed a new class of finite difference schemes to achieve this goal. First of all, a new nonlinear spectral-like weighted group velocity control (WGVC) scheme, centered around the concept of group velocity, is achieved to preserve the optimal order and optimize the spectral properties simultaneously by designing a new smoothness indicators for wave packets. Furthermore, by embedding the WGVC scheme into shock-capturing schemes, we further achieve the shock-capturing property without introducing any artificial parameters. The final schemes are validated by various numerical experiments and the results show that such schemes are order-preserving, spectral-property-optimized, and shock-capturing finite difference schemes and are very suitable for numerical simulation of compressible flow problems.