Unconditional Stability and Optimal Error Estimates of Euler Implicit/Explicit-SAV Scheme for the Navier-Stokes Equations

The unconditional stability and convergence analysis of the Euler implicit/explicit scheme with finite element discretization are studied for the incompressible time-dependent Navier-Stokes equations based on the scalar auxiliary variable approach. Firstly, a corresponding equivalent system of the Navier-Stokes equations with three variables is formulated, the stable finite element spaces are adopted to approximate these variables and the corresponding theoretical analysis results are provided. Secondly, a fully discrete scheme based on the backward Euler method is developed, the temporal treatment is based on the Euler implicit/explicit scheme, which is implicit for the linear terms and explicit for the nonlinear term. Hence, a constant coefficient algebraic system is formed and it can be solved efficiently. The discrete unconditional energy dissipation and stability of numerical solutions in various norms are established with any restriction on the time step, optimal error estimates are also provided. Finally, some numerical results are provided to illustrate the performances of the considered numerical scheme.

Automated summary

The study examines the unconditional stability and convergence of the Euler implicit/explicit scheme with finite element discretization for the incompressible time-dependent Navier-Stokes equations. By formulating an equivalent system and developing a discrete scheme, the implicit treatment of linear and explicit treatment of nonlinear terms is addressed, leading to an efficient solution method. The research establishes discrete unconditional energy dissipation and stability of numerical solutions without restrictions on the time step, along with optimal error estimates.