Seventh and ninth orders characteristic-wise alternative WENO finite difference schemes for hyperbolic conservation laws

In this work, the characteristic-wise alternative formulation of the seventh and ninth orders conservative weighted essentially non-oscillatory (AWENO) finite difference schemes is derived. The polynomial reconstruction procedure is applied to the conservative variables rather than the flux function of the classical WENO scheme. The numerical flux contains a low order term and high order derivative terms. The low order term can use arbitrary monotone fluxes that can enhance the resolution and reduce numerical dissipation of the fine scale structures while capturing shocks essentially non-oscillatory. The high order derivative terms are approximated by the central finite difference schemes. The improved performance in terms of accuracy, essentially non-oscillatory shock capturing and resolution for the complex shocked flow with fine scale structures in the classical one- and two-dimensional problems is demonstrated. However, the inclusion of the high order derivative terms is prone to generate Gibbs oscillations around a strong discontinuity and might result in a negative density and/or pressure. Therefore, a positivity-preserving limiter [Hu et al. J. Comput. Phys. 242 (2013)] is adopted to ensure the positive density and pressure in the shocked flows with extreme conditions, such as Mach 2000 jet flow problem.