A second order ensemble method with different subdomain time steps for simulating coupled surface-groundwater flows

In this article, we propose and study a second order, artificial compressibility ensemble method with different subdomain time steps for the Stokes-Darcy equations, which are the core model for simulating the coupled surface-groundwater flow ensembles. The algorithm decouples the system into two subphysics problems and allows larger time steps in the porous media region while using smaller time steps in the free flow region. We also adopt an artificial compressibility technique to further decouple the computation of the velocity and the pressure for which the pressure can be updated directly without solving a Poisson equation. The ensemble algorithm results in a common coefficient matrix for all realizations of different samples, consequently allows the use of efficient direct or iterative solvers to reduce the computational cost. We provide comprehensive stability analysis for both the nonensemble and the ensemble algorithms. Our numerical experiments illustrate the algorithm is second-order convergent and the ensemble scheme reduces the computational time of the nonensemble counterpart by 96%. A realistic numerical experiment on a curvy domain is also presented.