Highly efficient ensemble algorithms for computing the Stokes-Darcy equations

In this report we propose two highly efficient ensemble algorithms incorporating the gPAV and the rotational pressure correction methods for computing Stokes-Darcy flow ensembles. All variables are fully decoupled including the three components of the velocity, leading to smaller linear systems to be solved at each time step. Moreover, all ensemble members share the same constant coefficient matrix for which the fast block CG method can be applied for computing the ensemble at one pass at significantly reduced computational cost. We prove the two new ensemble algorithms are unconditionally stable with respect to the modified energy without any constraints on the uncertain parameters or the time steps. We will provide details of implementation and discuss how to efficiently solve the corresponding linear systems. Numerical examples are presented to show the efficiency and effectiveness of the algorithms.