In this paper we propose to revisit the notion of simple Riemann solver both in Lagrangian and Eulerian coordinates following the seminal work of Gallice ”Positive and entropy stable Godunov-type schemes for gas dynamics and MHD equations in Lagrangian or Eulerian coordinates” in Numer. Math., 94, 2003. We provide in this work the relation between the Eulerian and Lagrangian forms of systems of conservation laws in 1D. Then an approximate (simple) Lagrangian Riemann solver for gas dynamics is derived based on the notions of positivity preservation and entropy control. Its Eulerian counterpart is further deduced. Next we build the associated 1D first-order accurate cell-centered Lagrangian Godunov-type Finite Volume scheme and show numerically its behaviors on classical test cases. Then using the Lagrangian–Eulerian relationships, we derive and test the Eulerian Godunov-type Finite Volume scheme, which inherits by construction the properties of the Lagrangian solver in terms of positivity preservation and well-defined CFL condition. At last we extend this Eulerian scheme to arbitrary orders of accuracy using a Runge–Kutta time discretization, polynomial reconstruction and an a posteriori MOOD limiting strategy. Numerical tests are carried out to assess the robustness, accuracy, and essentially non-oscillatory properties of the numerical methods.

在本文中，我们建议根据 Gallice 的开创性工作重新审视拉格朗日和欧拉坐标中的简单黎曼求解器的概念，“气体动力学的正和熵稳定 Godunov 型方案以及拉格朗日或欧拉坐标中的 MHD 方程”在 Numer 中。Math., 94, 2003。我们在这项工作中提供了一维守恒定律系统的欧拉形式和拉格朗日形式之间的关系。然后基于正值守恒和熵控制的概念推导出一个近似（简单）的气体动力学拉格朗日黎曼求解器。进一步推导出它的欧拉对应物。接下来，我们构建相关的一维一阶精确以细胞为中心的拉格朗日戈杜诺夫型有限体积方案，并在经典测试用例上以数值方式显示其行为。然后使用拉格朗日-欧拉关系，我们推导出并测试了 Eulerian Godunov 型有限体积方案，该方案通过构造拉格朗日求解器在正性保持和明确定义的 CFL 条件方面继承了属性。最后，我们使用 Runge-Kutta 时间离散化、多项式重建和后验的情绪限制策略。进行数值测试以评估数值方法的稳健性、准确性和基本非振荡特性。