Unconditional H2$$ {H}^2 $$‐stability of the Euler implicit/explicit SAV‐based scheme for the 2D Navier–Stokes equations with smooth or nonsmooth initial data

Abstract

In this article, we propose ‐unconditional stable schemes for solving time‐dependent incompressible Navier–Stokes equations with smooth or nonsmooth initial data, , . The ‐stability analysis is established by leveraging the scalar auxiliary variable (SAV) approach. When dealing with nonsmooth initial data, we utilize a limited number of iteration of the semi‐implicit scheme followed by the SAV scheme. The overall efficiency is greatly enhanced due to the minimal computational cost of the semi‐implicit scheme and the explicit treatment of the nonlinear term within the SAV approach. The proposed schemes investigate two types of scalar auxiliary variables: the energy‐based variable and the exponential‐based variable. Rigorous proofs of the ‐unconditional stability of both schemes have been provided. Notice that both proposed numerical schemes enjoy unconditional long time stability for smooth and nonsmooth initial data when . Numerical experiments have been conducted to demonstrate the theoretical results.