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.

可压缩的Euler方程与引力源项相结合，获得了静水平衡状态，在此状态下，通量项的梯度可以由源项中的那些项精确平衡。当数值求解它们时，在离散级别上非常需要这种精确保持平衡的特性。在这项研究中，我们针对具有理想气体状态方程的系统，分别设计了简单的高阶均衡平衡差加权基本非振荡（WENO-R / I）方案，该方案分别基于WENO重构和内插程序。和规定的引力场。实现均衡特性的主要思想是以特殊方式重写源项，并通过将非线性WENO微分算子与均质化条件那些通量术语。可以用数学方法证明所提出的良好平衡方案，以精确地保持静水等温和多态平衡态，同时保持真正的高阶精度。而且，所产生的方案具有很高的计算效率，并且可以直接使用Lax-Friedrichs数值通量实施到现有的有限差分WENO代码中，以求解可压缩的Euler方程。最后但并非最不重要的一点是，建议的均衡框架也适用于一类运动的平衡状态。通过大量的一维和二维数值示例，从高阶精度，良好平衡性，