Abstract
We present an overview of a modern, efficient approach for uncoupling groundwater–surface water flows governed by the fully evolutionary Stokes–Darcy equations. Referred to as non-iterative partitioned methods, these algorithms treat the coupling terms explicitly and at each time level require only one Stokes and one Darcy sub-physics solve, thus taking advantage of existing solvers optimized for each sub-flow. This strategy often results in a time-step condition for stability. Furthermore, small problem parameters, specifically those related to the physical characteristics of the porous media domain, can render certain time-step conditions impractical. Despite these obstacles, researchers have made significant progress towards efficient, stable, and accurate partitioned methods. Herein, we provide a comprehensive survey and comparison of recent developments utilizing these non-iterative numerical schemes.
Highlights
Access to the clean freshwater is absolutely imperative for the continued survival of humankind.As a necessity for our agricultural, industrial and domestic practices, water constitutes an integral part of all civilizations
In [6], Saffman proposed a modification to the Beavers–Joseph coupling condition by dropping the porous media averaged velocity u p, based on observations that the term u p · τbi is negligible compared to the fluid velocity u · τbi
An attractive alternative to fully implicit, fully coupled discretization is exploiting information obtained in previous time steps to construct a non-iterative uncoupling scheme, which only need a single Stokes solve and a single Darcy solve at each time step
Summary
Access to the clean freshwater is absolutely imperative for the continued survival of humankind. Even natural processes may result in contaminated freshwater, as evident in the devastation of forests growing above coastal aquifers from salt-water intrusion Tracking these contaminants necessitates accurate numerical models for this coupled flow. Methods of high order, such as Crank–Nicolson–Leap Frog [29], second-order backward-differentiation with Gear’s extrapolation [30], and Adam–Moulton–Bashforth [30,31] These methods use explicit discretizations for the coupling terms, all are known to be long-time stable and optimally convergent uniformly in time (possibly under a small time-step constraint). With the addition of suitable stabilization terms, it is possible to further enhance the stability property, for instance, a stabilized Crank–Nicolson–Leap Frog, developed in [32,33], requires no time-step restriction for the long-time stability and convergence Another way for uncoupling groundwater–surface water systems is using splitting schemes.
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