A biological vesicle in fluid environment is described by a conservative Allen–Cahn type phase-field model and the incompressible Navier–Stokes equations. To accurately and efficiently solve this complex system, we present a totally decoupled, linear, and second-order time-accurate method based on a time-dependent auxiliary variable methodology. The time-discretized versions of energy stability and unique solvability are analytically proved. By using a simple and effective energy correction technique, the consistency between the original and modified energies is enhanced. The proposed numerical algorithm is simple to implement because we only need to separately solve linear elliptic equations. Various computational tests are conducted to verify the performance of the proposed numerical algorithm.

流体环境中的生物囊泡由保守的 Allen-Cahn 型相场模型和不可压缩的 Navier-Stokes 方程描述。为了准确有效地解决这个复杂系统，我们提出了一种基于时间相关辅助变量方法的完全解耦、线性和二阶时间精确方法。分析证明了能量稳定性和独特可解性的时间离散版本。通过使用简单有效的能量校正技术，增强了原始能量和修改能量之间的一致性。所提出的数值算法很容易实现，因为我们只需要单独求解线性椭圆方程。进行了各种计算测试以验证所提出的数值算法的性能。