Uncoupling evolutionary groundwater-surface water flows: stabilized mixed methods in both porous media and fluid regions

This paper considers the robust numerical methods for solving the time-dependent Stokes-Darcy multiphysics problem that can be implemented by use of existing surface water and groundwater codes. Porous media problem for the groundwater flow is preferable to employ the mixed discretization due to their superior conservation property and the convenience to compute flux on the large domain with relatively coarse meshes. However, the theory of mixed spatial discretizations for the time-dependent problems is far less developed than the non-mixed approaches, even for the one domain problems. Herein, we develop a stabilized mixed discretization technique for the porous media problem coupled with the fluid region across an interface with the physically appropriate coupling conditions. Time discretization is constructed to allow a non-iterative splitting of the coupled problem into two subproblems. The stability and convergence analysis of the coupled and decoupled algorithms are derived rigorously. If the time scale is bounded by a constant which only depends on the physical parameters, we prove the unconditional stability of both schemes. Four numerical experiments are conducted to show the efficiency and accuracy of the numerical methods, which illustrate the exclusive features of the Stokes-Darcy interface system.