可压缩欧拉方程的热力学相容半隐式有限体积格式

中文摘要 AI 辅助

在本工作中，我们为气体动力学的可压缩欧拉方程构建了一种新的半隐式（SI）双曲型且热力学相容（HTC）格式。其中的挑战在于将熵不等式作为主要演化方程来求解，而非总能量守恒定律，在我们的框架中，总能量守恒定律是一个附加的守恒律，需要通过相容的离散化由数值格式来满足。所提出的方法具有预测-校正特性，我们首先在不考虑熵产生的情况下求解等熵问题。在此过程中，熵和动量通量中的非线性对流项被显式离散，而质量通量和压力项则被隐式处理。随后，将离散动量方程代入离散质量守恒方程，得到一个关于压力的轻度非线性系统，该系统可通过嵌套的牛顿型算法高效求解。基于该预测，在熵中添加一个非负的产生项，从而确保数值格式满足单元熵不等式。为了在离散层面保证总能量的全局守恒，引入了一种新的修正的全局Abgrall型通量校正。所提出的数值方法允许的时间步长仅受物质波速限制，而不受声速限制。这使得该格式特别适用于低马赫数流动。文中展示了在一维和二维空间中的数值结果，以评估新格式的理论性质。

英文摘要

In this work we construct a new semi-implicit (SI) hyperbolic and thermodynamically compatible (HTC) scheme for the compressible Euler equations of gasdynamics. The challenge therein lies in solving the entropy inequality as the primary evolution equation instead of the total energy conservation law, which in our framework is an additional conservation law that needs to be fulfilled by the numerical scheme as the consequence of a compatible discretization. The proposed method has a predictor-corrector character, where we first solve the isentropic problem without considering the entropy production. Therein, the entropy and the nonlinear convective terms in the momentum flux are discretized explicitly, while the mass flux and the pressure terms are taken implicitly. The discrete momentum equation is then inserted into the discrete mass conservation equation, leading to a mildly nonlinear system for the pressure, which can be efficiently solved with a nested Newton-type algorithm. Based on this prediction, a non-negative production term in the entropy is added, ensuring a cell entropy inequality for the numerical scheme. To guarantee global conservation of total energy at the discrete level, a new modified global Abgrall-type flux correction is introduced. The presented numerical method allows time steps that are restricted only by the material wave speed and not by the sound speed. This makes the scheme particularly useful for flows in the low-Mach-number regime. Numerical results are shown in one and two space dimensions to assess the theoretical properties of the new scheme.

发表机构

• University of Trento（特伦托大学）
• Southern University of Science and Technology（南方科技大学）
• La Sapienza Università di Roma（罗马第一大学）
• Université de Strasbourg（斯特拉斯堡大学）
• Inria

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