This paper is focused on the approximation of the Euler equations of compressible fluid dynamics on a staggered mesh. With this aim, the flow parameters are described by the velocity, the density and the internal energy. The thermodynamic quantities are described on the elements of the mesh, and thus the approximation is only in $L^2$, while the kinematic quantities are globally continuous. The method is general in the sense that the thermodynamic and kinetic parameters are described by an arbitrary degree of polynomials. In practice, the difference between the degrees of the kinematic parameters and the thermodynamic ones {is set} to $1$. The integration in time is done using the forward Euler method but can be extended straightforwardly to higher-order methods. In order to guarantee that the limit solution will be a weak solution of the problem, we introduce a general correction method in the spirit of the Lagrangian staggered method described in \cite{Svetlana,MR4059382, MR3023731}, and we prove a Lax Wendroff theorem. The proof is valid for multidimensional versions of the scheme, even though most of the numerical illustrations in this work, on classical benchmark problems, are one-dimensional because we have easy access to the exact solution for comparison. We conclude by explaining that the method is general and can be used in different settings, for example, Finite Volume, or discontinuous Galerkin method, not just the specific one presented in this paper.

翻译：本文聚焦于交错网格上可压缩流体动力学欧拉方程的逼近。为此，流动参数通过速度、密度和内能进行描述。热力学量在网格单元上描述，因此近似仅在$L^2$空间中，而运动学量则是全局连续的。该方法具有通用性，热力学和动力学参数可由任意次多项式表示。实际应用中，运动学参数与热力学参数的阶次之差设定为$1$。时间积分采用前向欧拉方法，但可直接扩展至高阶方法。为保证极限解为问题的弱解，我们引入了一种基于文献\cite{Svetlana,MR4059382, MR3023731}中拉格朗日交错方法思想的通用修正技术，并证明了Lax-Wendroff定理。该证明适用于多维格式，尽管本文在经典基准问题上的数值算例多为一维情形——这主要是为了便于与精确解进行对比。最后我们指出，该方法具有通用性，可应用于有限体积法、间断伽辽金法等多种框架，而不仅限于本文介绍的具体方案。