We develop, and implement in a Finite Volume environment, a density-based approach for the Euler equations written in conservative form using density, momentum, and total energy as variables. Under simplifying assumptions, these equations are used to describe non-hydrostatic atmospheric flow. The well-balancing of the approach is ensured by a local hydrostatic reconstruction updated in runtime during the simulation to keep the numerical error under control. To approximate the solution of the Riemann problem, we consider four methods: Roe-Pike, HLLC, AUSM+-up and HLLC-AUSM. We assess our density-based approach and compare the accuracy of these four approximated Riemann solvers using two two classical benchmarks, namely the smooth rising thermal bubble and the density current.

翻译：我们在有限体积环境中开发并实现了一种基于密度的欧拉方程求解方法，该方程采用守恒形式，以密度、动量和总能作为变量。在简化假设下，这些方程用于描述非静力大气流动。该方法的平衡特性通过运行时更新的局部静力重构来保证，从而在模拟过程中将数值误差控制在合理范围内。为近似黎曼问题的解，我们考虑了四种方法：Roe-Pike、HLLC、AUSM+-up 和 HLLC-AUSM。我们评估了所提出的基于密度的求解方法，并利用两个经典基准测试案例（即光滑上升热泡和密度流）比较了这四种近似黎曼求解器的精度。