High order well-balanced positivity-preserving scale-invariant AWENO scheme for Euler systems with gravitational field

In this paper, we propose a fifth order well-balanced positivity-preserving finite difference scale-invariant AWENO scheme for the compressible Euler equations with gravitational fields. By using the scale-invariant WENO (Si-WENO) operator and well-balanced modification of the interpolated conservative variables, the finite difference discretization is well-balanced with respect to the priorly known isothermal and isentropic hydrostatic states. To ensure positivity of the density and pressure throughout the whole computation, we introduce interpolation-based and flux-based positivity-preserving limiters to both the density and pressure. Meanwhile, modifications are made to the discretization of the pressure equilibrium to restore well-balancedness. We point out that by using the Si-WENO operator we can compute all ingredients in the discretization of the source term prior to the time evolution, and the well-balanced and positivity-preserving modifications are made based on these ingredients, which can improve computational efficiency. Moreover, we carefully derive the positivity-preserving CFL conditions in one and two dimensions. Finally, the accuracy, robustness, effectiveness and numerical symmetry of our approach are demonstrated by a variety of numerical examples, where the time-marching strategy is used in two-dimensional problems to avoid strong dependence on p / ρ in the CFL conditions.