The present work concerns the derivation of a numerical scheme to approximate weak solutions of the Euler equations with a gravitational source term. The designed scheme is proved to be fully well-balanced since it is able to exactly preserve all moving equilibrium solutions, as well as the corresponding steady solutions at rest obtained when the velocity vanishes. Moreover, the proposed scheme is entropy-preserving since it satisfies all fully discrete entropy inequalities. In addition, in order to satisfy the required admissibility of the approximate solutions, the positivity of both approximate density and pressure is established. Several numerical experiments attest the relevance of the developed numerical method.

翻译：本研究致力于推导一种数值格式，用于近似求解含重力源项的欧拉方程的弱解。所设计的格式被证明具有完全保平衡性，因为它能够精确保持所有运动平衡解，以及在速度为零时对应的静止定常解。此外，该格式具有熵保持特性，满足所有全离散熵不等式。同时，为保证近似解的可容许性，建立了近似密度与压力正性的严格证明。若干数值实验验证了所发展数值方法的有效性。