We propose in this paper efficient first/second-order time-stepping schemes for the evolutional Navier-Stokes-Nernst-Planck-Poisson equations. The proposed schemes are constructed using an auxiliary variable reformulation and sophisticated treatment of the terms coupling different equations. By introducing a dynamic equation for the auxiliary variable and reformulating the original equations into an equivalent system, we construct first- and second-order semi-implicit linearized schemes for the underlying problem. The main advantages of the proposed method are: (1) the schemes are unconditionally stable in the sense that a discrete energy keeps decay during the time stepping; (2) the concentration components of the discrete solution preserve positivity and mass conservation; (3) the delicate implementation shows that the proposed schemes can be very efficiently realized, with computational complexity close to a semi-implicit scheme. Some numerical examples are presented to demonstrate the accuracy and performance of the proposed method. As far as the best we know, this is the first second-order method which satisfies all the above properties for the Navier-Stokes-Nernst-Planck-Poisson equations.

翻译：本文针对演化型Navier-Stokes-Nernst-Planck-Poisson方程组，提出了高效的一阶/二阶时间步进格式。所提格式基于辅助变量重构技术以及对耦合不同方程项的精细处理构建。通过引入辅助变量的动态方程并将原方程组重构为等价系统，我们为目标问题构建了一阶和二阶半隐式线性化格式。该方法的主要优势在于：(1) 格式具有无条件稳定性，即在时间步进过程中离散能量保持衰减特性；(2) 离散解中的浓度分量保持正性和质量守恒；(3) 精细实现表明，所提格式可实现极高效计算，其计算复杂度接近半隐式格式。数值算例展示了该方法的精度与性能。据我们所知，这是首个针对Navier-Stokes-Nernst-Planck-Poisson方程组同时满足上述所有性质的二阶方法。