High order well-balanced conservative finite difference AWENO scheme with hydrostatic reconstruction for the Euler equations under gravitational fields

The system of compressible Euler equations under gravitational fields drew much attention during the past few decades, due to its broad applications in astrophysics and atmospheric sciences. A critical feature of the system is that it admits a hydrostatic equilibrium state when the gradient of fluid flux is exactly balanced by the source term brought by gravitational fields, which is referred to as well-balancedness. In this paper, a well-balanced high order conservative finite difference alternative WENO (AWENO) scheme is proposed for the system. The finite difference AWENO scheme allows us to use an arbitrary consistent Riemann solver, such as LF, HLLC, etc. To preserve hydrostatic equilibrium, the hydrostatic reconstruction technique is adopted. We slightly modify the original hydrostatic reconstruction procedure to achieve the fifth order of accuracy in the framework of finite difference methods, without introducing spurious oscillations. Besides, the discretization is fully conservative without the source term, which is not the case for the methods with original hydrostatic reconstructions. Moreover, we will show that the well-balancedness is exactly guaranteed by our approach at the discrete level. Finally, numerical examples illustrate the effectiveness and robustness of the designed approach.