Stabilized Scalar Auxiliary Variable Ensemble Algorithms for Parameterized Flow Problems

Abstract

Computing a flow system a number of times with different samples of flow parameters is a common practice in many uncertainty quantification applications, which can be prohibitively expensive for complex nonlinear flow problems. This report presents two second order, stabilized, scalar auxiliary variable (SAV) ensemble algorithms for fast computation of the Navier--Stokes flow ensembles: Stab-SAV-CN and Stab-SAV-BDF2. The proposed ensemble algorithms are based on the ensemble timestepping idea which makes use of a quantity called the ensemble mean to construct a common coefficient matrix for all realizations at the same time step after spatial discretization, in which case efficient block solvers, e.g., block GMRES, can be used to significantly reduce both storage and computational time. The adoption of a recently developed SAV approach that treats the nonlinear term explicitly results in a constant shared coefficient matrix among all realizations at different time steps, which further cuts down the computational cost, yielding an extremely efficient ensemble algorithm for simulating nonlinear flow ensembles with provable long time stability without any time step conditions. The SAV approach for the Navier--Stokes equations for a single realization was proved to be unconditionally stable in [L. Lin, Z. Yang, and S. Dong, J. Comput. Phys., 388 (2019), pp 1--22; X. Li and J. Shen, SIAM J. Numer. Anal., 58 (2020), pp. 2465--2491]. However we found the SAV approach has very low accuracy that compromises its stability in our initial numerical investigations for several commonly tested benchmark flow problems. In this report, we propose to use the stabilization $-\alpha h \Delta (u^{n+1}-u^n)$ in Stab-SAV-CN and $-\alpha h \Delta (3u^{n+1}-4 u^{n}+u^{n-1})$ in Stab-SAV-BDF2 to address this issue. We prove that both of our ensemble algorithms are long time stable under one parameter fluctuation condition, without any time step constraints. For a single realization, both algorithms are unconditionally stable and have better accuracy than the SAV methods studied in [L. Lin, Z. Yang, and S. Dong, J. Comput. Phys., 388 (2019), pp 1--22; X. Li and J. Shen, SIAM J. Numer. Anal., 58 (2020), pp. 2465--2491] for our test problems. Extensive numerical experiments are performed to show the efficiency of the proposed ensemble algorithms and the effectiveness of the stabilization for increasing accuracy and stability.