Numerical investigation of two second-order, stabilized SAV ensemble methods for the Navier-Stokes equations

In this report we present a second-order, stabilized SAV based, Crank-Nicolson leap-frog (CNLF) ensemble method, and perform a comprehensive numerical study of it as well as the Crank-Nicolson ensemble method with a linear extrapolation (CNLE) presented in Jiang and Yang (SIAM J. Sci. Comput. 43:A2869-A2896, 2021). Both methods are extremely efficient as only one linear system with multiple right hands needs to be solved at each time for a (potentially large) number of realizations of the flow problems. In particular the coefficient matrix of the fully discretized system is a constant matrix that does not change from one time step to another. We present extensive testing of these two methods and demonstrate the advantages of each. We also present long time stability analysis for both methods.