Partitioned Time Stepping Method for Fully Evolutionary Stokes--Darcy Flow with Beavers--Joseph Interface Conditions

In this report, a partitioned time stepping algorithm for transient flow in a matrix domain coupled to a free flow in embedded conduits is analyzed. The coupled flow is modeled by the fully evolutionary Stokes--Darcy problem. This method requires solving only one, uncoupled Stokes and Darcy subphysics and subdomain per time step. On the interface between the matrix and conduit, Beavers--Joseph interface conditions, instead of the simplified Beavers--Joseph--Saffman condition, are imposed. Under a modest time step restriction of the form $\triangle t\leq C$, where $C=C$ (physical parameters) we prove stability of the method. We also derive error estimates. Numerical tests illustrate the validity of the theoretical results.