In this paper, we construct and analyze new first- and second-order implicit-explicit (IMEX) schemes for the unsteady Navier-Stokes-Darcy model to describe the coupled free flow-porous media system, which is based on the scalar auxiliary variable (SAV) approach in time and finite element method in space. The constructed schemes are linear, only require solving a sequence of linear differential equations with constant coefficients at each time step, and can decouple the Navier-Stokes and Darcy systems. The unconditional stability of both the first- and second-order IMEX schemes can be derived for the coupled system equipped with the Lions interface condition, where the key point is that we should construct a new trilinear form to balance the fully explicit discretizations of the nonlinear terms in the complex system. We can also establish rigorous error estimates for the velocity and hydraulic head of the first-order scheme without any time step restriction. Numerical examples are presented to validate the proposed schemes.

翻译：本文针对非定常Navier-Stokes-Darcy模型（描述耦合自由流-多孔介质系统），基于时间方向上的标量辅助变量（SAV）方法和空间方向上的有限元方法，构建并分析了新的一阶和二阶隐式-显式（IMEX）格式。所构建的格式具有线性特性，每个时间步仅需求解一系列常系数线性微分方程，并可解耦Navier-Stokes方程与Darcy系统。对于配备Lions界面条件的耦合系统，可推导出一阶和二阶IMEX格式的无条件稳定性，其关键在于需构造一种新的三线性形式来平衡复杂系统中非线性项的完全显式离散。我们还可针对一阶格式的速度和水头建立严格的误差估计，且无需任何时间步长限制。数值算例验证了所提格式的有效性。