This work focuses on modeling the interaction between an incompressible, viscous fluid and a poroviscoelastic material. The fluid flow is described using the time‐dependent Stokes equations, and the poroelastic material using the Biot model. The viscoelasticity is incorporated in the equations using a linear Kelvin–Voigt model. We introduce two novel, noniterative, partitioned numerical schemes for the coupled problem. The first method uses the second‐order backward differentiation formula (BDF2) for implicit integration, while treating the interface terms explicitly using a second‐order extrapolation formula. The second method is the Crank–Nicolson and Leap‐Frog (CNLF) method, where the Crank–Nicolson method is used to implicitly advance the solution in time, while the coupling terms are explicitly approximated by the Leap‐Frog integration. We show that the BDF2 method is unconditionally stable and uniformly stable in time, while the CNLF method is stable under a CFL condition. Both schemes are validated using numerical simulations. Second‐order convergence in time is observed for both methods. Simulations over a longer period of time show that the errors in the solution remain bounded. Cases when the structure is poroviscoelastic and poroelastic are included in numerical examples.

中文翻译：

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流体-多孔弹性材料相互作用的二阶松耦合方法

这项工作着重于对不可压缩的粘性流体和多孔黏弹性材料之间的相互作用进行建模。使用与时间有关的斯托克斯方程来描述流体流动，并使用毕奥特模型来描述多孔弹性材料。使用线性Kelvin-Voigt模型将粘弹性纳入方程式中。对于耦合问题，我们介绍了两种新颖的，非迭代的，分区数值方案。第一种方法使用二阶后向微分公式（BDF2）进行隐式积分，同时使用二阶外推公式显式处理接口项。第二种方法是Crank-Nicolson和Leap-Frog（CNLF）方法，其中使用Crank-Nicolson方法隐式地提前求解，而耦合项由Leap-Frog积分显式近似。我们表明，BDF2方法是无条件稳定且时间上均匀稳定的，而CNLF方法在CFL条件下是稳定的。两种方案都使用数值模拟进行了验证。两种方法都可以观察到时间的二阶收敛。较长时间的仿真表明，解决方案中的错误仍然有限。数值示例包括结构为多孔弹性和多孔弹性的情况。