SAV decoupled ensemble algorithms for fast computation of Stokes–Darcy flow ensembles

Abstract

Numerical modeling and simulation of complex systems is often subject to uncertainties in model parameters. Many popular uncertainty quantification (UQ) methods require repeated simulations of the underlying physical system with different samples of the uncertain model parameters. This poses great challenges to many practical engineering applications due to the high demand for computational resources. In this report we propose highly efficient ensemble simulation algorithms for fast computation of coupled flow ensembles. The proposed ensemble algorithms are based on two recently developed numerical approaches: scalar auxiliary variable (SAV) and ensemble timestepping. We introduce a new decoupling strategy using the SAV idea and incorporate the ensemble timestepping method to develop two decoupled ensemble schemes for the Stokes–Darcy system: SAV-BE-En and SAV-BDF2-En. The two ensemble algorithms are specially designed for UQ computations where a number of realizations of the underlying coupled PDE system are required for analyzing and interpreting flow statistics. Compared with traditional methods which solve for each realization independently, our proposed ensemble algorithms result in a common coefficient matrix for all realizations and efficient iterative solvers such as block CG or block GMRES can be used to solve for all realizations simultaneously reducing both computer storage and overall simulation time. We prove that both ensemble algorithms are long time stable without any time step conditions. We also provide a comprehensive error analysis for the fully discrete SAV-BE-En algorithm, and present a few illustrative numerical examples to demonstrate the efficiency and effectiveness of the algorithms.