Second-order, loosely coupled methods for fluid-poroelastic material interaction

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.