基于熵的选择原理在开尔文-亥姆霍兹不稳定性实例中的数值实现

中文摘要 AI 辅助

本文我们将基于参数化Young测度和线性规划的数值格式推广到二维空间，并将其应用于由气体动力学二维欧拉方程控制的开尔文-亥姆霍兹不稳定性。我们构造了一阶、二阶、三阶、五阶、七阶和九阶Young测度格式，并将它们与相应的标准局部Lax-Friedrichs（LLF）通量分裂格式以及配备基于熵的局部特征分解的LLF格式（LLF-ELCD）进行比较。我们考察了瞬时和时均密度剖面、跨空间重构阶数的累积平均值、区域经验密度分布、平均密度边际分布，以及耗散弱解的若干选择准则。数值结果揭示了依赖于格式的流动模式，特别是在剪切层卷起过程中产生的小尺度结构。在所考虑的每个重构阶数下，Young测度格式产生最大的时均物理熵。相比之下，LLF-ELCD格式并不系统地产生比标准LLF格式更大的熵，因此不构成一致的最大熵选择机制。Young测度的平均密度边际分布集中在几个相邻的密度状态上，并且对于所有Young测度格式，后期的累积平均值更接近。这些结果证明了二维Young测度公式的可行性，并表明线性规划问题中的目标函数可以充当有效的选择机制。观察到的熵偏好与预期物理熵的局部优化一致。仅通过将熵纳入传统数值构造中无法重现这一现象。

英文摘要

In this paper, we extend numerical schemes based on parameterized Young measures and linear programming to two space dimensions and apply them to the Kelvin-Helmholtz instability governed by the two-dimensional Euler equations of gas dynamics. We construct first-, second-, third-, fifth-, seventh-, and ninth-order Young-measure schemes and compare them with corresponding standard local Lax-Friedrichs (LLF) flux-splitting schemes and LLF schemes equipped with an entropy-based local characteristic decomposition (LLF-ELCD). We examine instantaneous and time-averaged density profiles, cumulative averages across spatial reconstruction orders, regional empirical density distributions, averaged density marginals, and several selection criteria for dissipative weak solutions. The numerical results reveal scheme-dependent flow patterns, particularly in the small-scale structures generated during the roll-up of the shear layers. At every reconstruction order considered, the Young-measure schemes yield the largest time-averaged physical entropy. By contrast, the LLF-ELCD schemes do not systematically yield larger entropy than the standard LLF schemes and therefore do not constitute a consistent maximum-entropy selection mechanism. The averaged density marginals of the Young measures are concentrated on several neighboring density states, and the later cumulative averages are closer for all Young-measure schemes. These results demonstrate the feasibility of the two-dimensional Young-measure formulation and show that the objective function in the linear-programming problem can act as an effective selection mechanism. The observed entropy preference is consistent with the local optimization of the expected physical entropy. It cannot be reproduced by merely incorporating entropy into a conventional numerical construction.

发表机构

• RWTH Aachen University（亚琛工业大学）
• University of Pretoria（比勒陀利亚大学）

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