A decoupling method with different subdomain time steps for the nonstationary stokes–darcy model

Abstract

This report analyzes a multirate, decoupling algorithm, which allows different time steps in the fluid region and the porous region for the nonstationary Stokes–Darcy problem. The method presented requires only one, uncoupled Stokes and Darcy subphysics and subdomain solve per time step. Under a time step restriction of the form △t ≤ C (physical parameters) we prove stability and convergence of the method. Numerical tests are given to show the convergence result and demonstrate the computational efficiency of the partitioned method. They also show that in the expected case of greater fluid velocities in the free‐flow region than in the porous media region, allowing smaller time steps in the subregion with the faster velocities increases both accuracy and efficiency. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Non-Iterative Partitioned Methods for Uncoupling Evolutionary Groundwater–Surface Water Flows

“…The stability of BEFE-as1 was proved in [35] over a bounded interval. In particular, one has the following: …”

“…As the interface values are obtained in an explicit manner, the main challenge here is how to obtain optimal accuracy and good stability properties. Many non-iterative partitioned methods have been developed in the literature recently [27][28][29][30][31][32][33][34][35][36]42], whose stability and accuracy have been proved (over a long time or without time-step condition) and numerically tested. Several of them maintain good performance even in the case of small parameters.…”

“…In such cases, it may be desirable to apply an asynchronous scheme (aka, multiple-time-step scheme, multi-rate scheme), which computes fast solutions using a small time step and consider a larger time step for slow solutions. The first partitioned scheme that allows different time steps in the fluid and porous region for the nonstationary Stokes-Darcy problem was probably proposed and analyzed in [35]. In that work, the decoupling is based on lagging the interfacial coupling terms following the BEFE method; thus, we will refer to the scheme as asynchronous BEFE or BEFE-as1.…”

“…In addition, define n k := kr and let N = T/∆s, the number of small time step, and M = T/∆t = N/r, the number of large time step. The algorithm in [35] reads as follows.…”

“…Such methods may be more efficient, as we may apply two different time steps to separately solve the fast and slow flows. Developed in [35,36], these asynchronous techniques utilize the Backward Euler-Forward Euler time discretization, with long-time stability acquired in the latter work.…”