Thermodynamically consistent models in continuum physics, i.e. models which satisfy the first and second laws of thermodynamics, may be expressed using the metriplectic formalism. In this work, we leverage the structures underlying this modeling formalism to preserve thermodynamic consistency in discretizations of a fluid model. The procedure relies (1) on ensuring that the spatial semi-discretization retains certain symmetries and degeneracies of the Poisson and metriplectic 4-brackets, and (2) on the use of an appropriate energy conserving time-stepping method. The minimally simple yet nontrivial example of a one-dimensional thermal-fluid model is treated. It is found that preservation of the requisite symmetries and degeneracies of the 4-bracket is relatively simple to ensure in Galerkin spatial discretizations, suggesting a path forward for thermodynamically consistent discretizations of more complex fluid models using more specialized Galerkin methods.

中文翻译：

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使用其三重 4 支架结构的一维热流体模型的热力学一致离散化


连续介质物理学中的热力学一致模型，即满足热力学第一定律和第二定律的模型，可以用三重形式表示。在这项工作中，我们利用这种建模形式的基础结构来保持流体模型离散化中的热力学一致性。该过程依赖于 （1） 确保空间半离散化保留泊松和中三重 4 括号的某些对称性和简并性，以及 （2） 使用适当的能量守恒时间步进方法。本文讨论了一维热流体模型的最小简单但重要的示例。研究发现，在 Galerkin 空间离散化中确保 4 括号的必要对称性和简并性相对简单，这表明使用更专业的 Galerkin 方法对更复杂的流体模型进行热力学一致离散化提供了一条前进的道路。