基于总压力的耦合斯托克斯-比奥模型的二阶单块格式
A Second-Order Monolithic Scheme for the Coupled Stokes--Biot Model Using Total Pressure
AI总结：
针对耦合斯托克斯-比奥模型，本文开发了一种基于总压力的二阶全隐式单块时间离散格式，证明其能量稳定性与收敛性，数值实验验证了该格式的二阶时间精度。
AI中文摘要：
本文针对耦合的随时间变化的斯托克斯系统与准静态比奥系统，开发并分析了一种二阶、全隐式、单块的时间离散格式。比原子系统采用三场总压力形式，其中总压力作为额外未知量，与固体位移、孔隙压力一同引入；该重构提升了近不可压缩区域的鲁棒性，抑制了体积闭锁。所有变量均采用BDF2格式进行时间离散，耦合问题无需算子拆分即可单块求解。利用BDF2的G-稳定性恒等式建立了离散能量稳定性，所得估计对拉梅参数、比奥-威利斯系数及存储系数均具有鲁棒性。随后采用相容性-稳定性论证推导了时间二阶收敛性及空间最优阶收敛性。数值实验验证了理论结果，证实了对应能量范数下的时间二阶精度。
英文摘要：
We develop and analyze a second-order, fully implicit, monolithic time-discretization for the coupled time-dependent Stokes and quasi-static Biot system. The Biot subsystem is formulated in a three-field total-pressure formulation, in which the total pressure is introduced as an additional unknown together with the solid displacement and pore pressure. This reformulation improves robustness in nearly incompressible regimes and suppresses volumetric locking. All variables are discretized in time using the BDF2 scheme, and the coupled problem is solved monolithically without operator splitting. Discrete energy stability is established using the BDF2 $G$-stability identity. The resulting estimates exhibit robustness with respect to the Lamé parameter, Biot--Willis coefficient, and storage coefficient. A consistency--stability argument is then used to derive second-order convergence in time together with optimal-order convergence in space. Numerical experiments confirm the theoretical results and demonstrate second-order temporal accuracy in the corresponding energy norms.
