COUPLED SURFACE AND GROUNDWATER FLOWS: QUASISTATIC LIMIT AND A SECOND-ORDER, UNCONDITIONALLY STABLE, PARTITIONED METHOD

In this thesis we study the fully evolutionary Stokes-Darcy and Navier-Stokes/Darcy models for the coupling of surface and groundwater flows versus the quasistatic models, in which the groundwater flow is assumed to instantaneously adjust to equilibrium. Further, we develop and analyze an efficient numerical method for the Stokes-Darcy problem that decouples the sub-physics flows, and is 2nd-order convergent, uniformly in the model parameters. \ud \ud We first investigate the linear, fully evolutionary Stokes-Darcy problem and its qua- sistatic approximation, and prove that the solution of the former converges to the solution of the latter as the specific storage parameter converges to zero. The proof reveals that the quasistatic problem predicts the solution accurately only under certain parameter regimes. \ud \ud Next, we develop and analyze a partitioned numerical method for the evolutionary Stokes- Darcy problem. We prove that the new method is asymptotically stable, and second-order, uniformly convergent with respect to the model parameters. As a result, it can be used to solve the quasistatic Stokes-Darcy problem. Several numerical tests are performed to support the theoretical efficiency, stability, and convergence properties of the proposed method. \ud \ud Finally, we consider the nonlinear Navier-Stokes/Darcy problem and its quasistatic ap- proximation under a modified balance of forces interface condition. We show that the solution of the fully evolutionary problem converges to the quasistatic solution as the specific stor- age converges to zero. To prove convergence in three spatial dimensions, we assume more regularity on the solution, or small data