Decoupling the Stationary Navier-Stokes-Darcy System with the Beavers-Joseph-Saffman Interface Condition

This paper proposes a domain decomposition method for the coupled stationary Navier-Stokes and Darcy equations with the Beavers-Joseph-Saffman interface condition in order to improve the efficiency of the finite element method. The physical interface conditions are directly utilized to construct the boundary conditions on the interface and then decouple the Navier-Stokes and Darcy equations. Newton iteration will be used to deal with the nonlinear systems. Numerical results are presented to illustrate the features of the proposed method

Long time stability of four methods for splitting the evolutionary Stokes–Darcy problem into Stokes and Darcy subproblems

AbstractThis report analyzes the long time stability of four methods for non-iterative, sub-physics, uncoupling for the evolutionary Stokes–Darcy problem. The four methods uncouple each timestep into separate Stokes and Darcy solves using ideas from splitting methods. Three methods uncouple sequentially while one is a parallel uncoupling method. We prove long time stability of four splitting based partitioned methods under timestep restrictions depending on the problem parameters. The methods include those that are stable uniformly in S0, the storativity coefficient, for moderate kmin, the minimum hydraulic conductivity, uniformly in kmin for moderate S0 and with no coupling between the timestep and the spacial meshwidth

Long time stability of four methods for splitting the evolutionary Stokes–Darcy problem into Stokes and Darcy subproblems

We dedicate this paper to Professor Jan Verwer. A . This report analyzes the long time stability of four methods for non-iterative, sub-physics, uncoupling for the evolutionary Stokes-Darcy problem. The four methods uncouple each time step into separate Stokes and Darcy solves using ideas from splitting methods. Three methods uncouple sequentially while one is a parallel uncoupling method. We prove long time stability of four splitting based partitioned methods under time step restrictions depending on the problem parameters. The methods include ones stable uniformly in S 0 , the storativity coefficient, for moderate k min , the minimum hydraulic conductivity, uniformly in k min for moderate S 0 and with no coupling between the timestep and the spacial meshwidth

Higher-Order, Strongly Stable Methods for Uncoupling Groundwater-Surface Water Flow

Many environmental problems today involve the prediction of the migration of contaminants in groundwater-surface water flow. Sources of contaminated groundwater-surface water flow include: landfill leachate, radioactive waste from underground storage containers, and chemical run-off from pesticide usage in agriculture, to name a few. Before we can track the transport of pollutants in environmental flow, we must first model the flow itself, which takes place in a variety of physical settings. This necessitates the development of accurate numerical models describing coupled fluid (surface water) and porous media (groundwater) flow, which we assume to be described by the fully evolutionary Stokes-Darcy equations. Difficulties include finding methods that converge within a reasonable amount of time, are stable when the physical parameters of the flow are small, and maintain stability and accuracy along the interface. Ideally, because there exist a wide variety of physical scenarios for this coupled flow, we desire numerical methods that are versatile in terms of stability and practical in terms of computational cost and time. The approach to model this flow studied herein seeks to take advantage of existing efficient solvers for the separate sub-flows by uncoupling the flow so that at each time level we may solve a separate surface and groundwater problem. This approach requires only one (SPD) Stokes and one (SPD) Darcy sub-physics and sub-domain solve per time level for the time-dependent Stokes-Darcy problem. In this dissertation, we investigate several different methods that uncouple groundwater-surface water flow, and provide thorough analysis of the stability and convergence of each method along with numerical experiments

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. 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. 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. 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