Multiscale Modeling &Simulation, Volume 18, Issue 1, Page 502-541, January 2020.
Well-balanced and free energy dissipative first- and second-order accurate finite-volume schemes are proposed for a general class of hydrodynamic systems with linear and nonlinear damping. The variation of the natural Lyapunov functional of the system, given by its free energy, allows for a characterization of the stationary states by its variation. An analogous property at the discrete level enables us to preserve stationary states at machine precision while keeping the dissipation of the discrete free energy. Performing a careful validation in a battery of relevant test cases, we show that these schemes can accurately analyze the stability properties of stationary states in challenging problems such as phase transitions in collective behavior, generalized Euler--Poisson systems in chemotaxis and astrophysics, and models in dynamic density functional theories.

中文翻译：

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具有一般自由能的流体动力学方程的均衡有限体积格式

多尺度建模与仿真，第18卷，第1期，第502-541页，2020年1月。
针对一类具有线性和非线性阻尼的流体力学系统，提出了一种具有良好平衡性和自由能耗散的一阶和二阶精确有限体积方案。由其自由能给出的系统自然Lyapunov功能的变化允许通过其变化来表征稳态。离散级的类似属性使我们能够以机器精度保留稳态，同时保持离散自由能的耗散。在一系列相关的测试案例中进行了仔细的验证，我们证明了这些方案可以准确地分析稳态问题的稳定性，这些挑战性问题包括集体行为的相变，趋化性的广义欧拉-泊松系统以及天体物理学，