Simple high order well-balanced finite difference WENO schemes for the Euler equations under gravitational fields

The compressible Euler equations coupled with the gravitational source terms admit a hydrostatic equilibrium state where the gradients of the flux terms can be exactly balanced by those in the source terms. This property of exact preservation of the equilibrium is highly desirable at the discrete level when they are numerically solved. In this study, we design the simple high order well-balanced finite difference weighted essentially non-oscillatory (WENO-R/I) schemes, which base on the WENO reconstruction and interpolation procedures respectively, for this system with an ideal gas equation of state and a prescribed gravitational field. The main idea in achieving the well-balanced property is to rewrite the source terms in a special way and discretize them by using the nonlinear WENO differential operators with the homogenization condition to those for the flux terms. The proposed well-balanced schemes can be proved mathematically to preserve the hydrostatic isothermal and polytropic equilibria states exactly and at the same time maintain genuine high order accuracy. Moreover, the resulting schemes are high efficiency of computation and can be implemented straightforwardly into the existing finite difference WENO code with the Lax-Friedrichs numerical flux for solving the compressible Euler equations. Last but not least, the proposed well-balanced framework also works for a class of moving equilibrium state. Extensive one- and two-dimensional numerical examples are carried out to investigate the performance of the proposed schemes in term of high order accuracy, well-balanced property, shock capturing essentially non-oscillatory and resolving the small perturbation on the coarse mesh resolution.