In this paper, we propose a linear, decoupled, unconditionally stable fully-discrete finite element scheme for the active fluid model, which is derived from the gradient flow approach for an effective non-equilibrium free energy. The developed scheme is employed by an implicit-explicit treatment of the nonlinear terms and a second-order Gauge–Uzawa method for the decoupling of computations for the velocity and pressure. We rigorously prove the unique solvability and unconditional stability of the proposed scheme. Several numerical tests are presented to verify the accuracy, stability, and efficiency of the proposed scheme. We also simulate the self-organized motion under the various external body forces in 2D and 3D cases, including the motion direction of active fluid from disorder to order. Numerical results show that the scheme has a good performance in accurately capturing and handling the complex dynamics of active fluid motion.

中文翻译：

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主动流体模型无条件稳定全离散有限元数值格式

在本文中，我们提出了一种用于主动流体模型的线性、解耦、无条件稳定的全离散有限元方案，该方案源自有效非平衡自由能的梯度流方法。所开发的方案通过非线性项的隐式-显式处理和二阶 Gauge-Uzawa 方法来解耦速度和压力的计算。我们严格证明了所提出方案的独特可解性和无条件稳定性。进行了多次数值试验来验证所提出方案的准确性、稳定性和效率。我们还模拟了 2D 和 3D 情况下各种外部体力下的自组织运动，包括活性流体从无序到有序的运动方向。数值结果表明，该方案在准确捕捉和处理主动流体运动的复杂动力学方面具有良好的性能。