推导可逆-不可逆热力学相容模型的线性、解耦和能量稳定格式的一般框架：第一部分不可压流体动力学模型
A General Framework to Derive Linear, Decoupled and Energy-stable Schemes for Reversible-Irreversible Thermodynamically Consistent Models: Part I Incompressible Hydrodynamic Models
摘要：
在本文中，我们提出了一个通用的数值平台，设计精确，高效，稳定的数值算法的不可压缩流体力学模型，遵守的经验法则。所得到的数值格式在时间上是自动线性的。该方法将水动力变量和其他状态变量进行叠加，使得在每一步时间推进中只需求解小规模的线性问题。如果采用经典的速度投影方法，可以实现速度场和压力场的解耦。最后，在每个时间步中只需要求解几个椭圆型方程。这一策略是通过充分探索模型的热力学结构，通过一系列的模型重新表述成为可能。广义Onsager原则直接指导这些重新制定程序。在重新表述但等效的模型中，可以识别可逆和不可逆分量，指导数值平台解耦可逆和不可逆动力学。这最终导致解耦的数值算法，因为耦合项只涉及不可逆动力学。为了进一步证明数值平台的力量，我们将其应用到几个特定的不可压缩流体动力学模型。文中详细讨论了所提出的数值格式的能量稳定性。时间上的二阶精度通过时间步长细化测试进行了数值验证。几个基准的数值例子，以进一步说明所提出的数值框架的准确性，稳定性和效率。
英文摘要：
In this paper, we present a general numerical platform for designing accurate, efficient, and stable numerical algorithms for incompressible hydrodynamic models that obeys the thermodynamical laws. The obtained numerical schemes are automatically linear in time. It decouples the hydrodynamic variable and other state variables such that only small-size linear problems need to be solved at each time marching step. Furthermore, if the classical velocity projection method is utilized, the velocity field and pressure field can be decoupled. In the end, only a few elliptic-type equations shall be solved in each time step. This strategy is made possible through a sequence of model reformulations by fully exploring the models' thermodynamic structures. The generalized Onsager principle directly guides these reformulation procedures. In the reformulated but equivalent models, the reversible and irreversible components can be identified, guiding the numerical platform to decouple the reversible and irreversible dynamics. This eventually leads to decoupled numerical algorithms, given that the coupling terms only involve irreversible dynamics. To further demonstrate the numerical platform's power, we apply it to several specific incompressible hydrodynamic models. The energy stability of the proposed numerical schemes is shown in detail. The second-order accuracy in time is verified numerically through time step refinement tests. Several benchmark numerical examples are presented to further illustrate the proposed numerical framework's accuracy, stability, and efficiency.
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