In this work we analyze the entropic properties of the Euler equations when the system is closed with the assumption of a polytropic gas. In this case, the pressure solely depends upon the density of the fluid and the energy equation is not necessary anymore as the mass conservation and momentum conservation then form a closed system. Further, the total energy acts as a convex mathematical entropy function for the polytropic Euler equations. The polytropic equation of state gives the pressure as a scaled power law of the density in terms of the adiabatic index $$\gamma $$ γ . As such, there are important limiting cases contained within the polytropic model like the isothermal Euler equations ( $$\gamma {=}1$$ γ = 1 ) and the shallow water equations ( $$\gamma {=}2$$ γ = 2 ). We first mimic the continuous entropy analysis on the discrete level in a finite volume context to get special numerical flux functions. Next, these numerical fluxes are incorporated into a particular discontinuous Galerkin (DG) spectral element framework where derivatives are approximated with summation-by-parts operators. This guarantees a high-order accurate DG numerical approximation to the polytropic Euler equations that is also consistent to its auxiliary total energy behavior. Numerical examples are provided to verify the theoretical derivations, i.e., the entropic properties of the high order DG scheme.

中文翻译：

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等温和多变欧拉方程的熵稳定数值近似

在这项工作中，我们分析了当系统在多方气体假设下封闭时欧拉方程的熵特性。在这种情况下，压力仅取决于流体的密度并且能量方程不再必要，因为质量守恒和动量守恒形成了一个封闭系统。此外，总能量充当多方欧拉方程的凸数学熵函数。多方状态方程根据绝热指数 $$\gamma $$ γ 将压力作为密度的缩放幂定律给出。因此，多方模型中包含重要的极限情况，如等温欧拉方程 ( $$\gamma {=}1$$ γ = 1 ) 和浅水方程 ( $$\gamma {=}2$$ γ = 2)。我们首先在有限体积上下文中模拟离散水平上的连续熵分析，以获得特殊的数值通量函数。接下来，这些数值通量被合并到一个特定的不连续伽辽金 (DG) 光谱元素框架中，其中导数用按部分求和算子近似。这保证了多方欧拉方程的高阶精确 DG 数值近似，这也与其辅助总能量行为一致。提供了数值例子来验证理论推导，即高阶 DG 方案的熵特性。这些数值通量被合并到一个特定的不连续伽辽金 (DG) 光谱元素框架中，其中导数用分部分求和算子近似。这保证了多方欧拉方程的高阶精确 DG 数值近似，这也与其辅助总能量行为一致。提供了数值例子来验证理论推导，即高阶 DG 方案的熵特性。这些数值通量被合并到一个特定的不连续伽辽金 (DG) 光谱元素框架中，其中导数用分部分求和算子近似。这保证了多方欧拉方程的高阶精确 DG 数值近似，这也与其辅助总能量行为一致。提供了数值例子来验证理论推导，即高阶 DG 方案的熵特性。