A class of new linear, efficient and high-order implicit-explicit methods for the coupled free flow-porous media system based on nonlinear Lions interface condition

TL;DR

This paper introduces new linear, high-order implicit-explicit schemes for the coupled free flow-porous media system, ensuring stability, decoupling, and rigorous error estimates using SAV and finite element methods.

Contribution

The paper develops and analyzes linear, high-order IMEX schemes based on SAV and finite element methods for the Navier-Stokes-Darcy model, with unconditional stability and decoupling.

Findings

Unconditionally stable first- and second-order schemes.

Decoupling of Navier-Stokes and Darcy systems.

Rigorous error estimates without time step restrictions.

Abstract

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.