An analysis of a linearly extrapolated BDF2 subgrid artificial viscosity method for incompressible flows

TL;DR

This paper extends the mathematical analysis of a subgrid artificial viscosity method combined with linearly extrapolated BDF2 time discretization for incompressible Navier-Stokes equations, demonstrating stability and convergence.

Contribution

It provides a detailed theoretical foundation and stability proof for a SAV method with BDF2LE discretization applied to incompressible flows.

Findings

Unconditionally stable solutions for the method.

Optimal convergence of the numerical solutions.

Validation through numerical tests confirming theoretical results.

Abstract

This report extends the mathematical support of a subgrid artificial viscosity (SAV) method to simulate the incompressible Navier-Stokes equations to better performing a linearly extrapolated BDF2 (BDF2LE) time discretization. The method considers the viscous term as a combination of the vorticity and the grad-div stabilization term. SAV method introduces global stabilization by adding a term, then antidiffuses through the extra mixed variables. We present a detailed analysis of conservation laws, including both energy and helicity balance of the method. We also show that the approximate solutions of the method are unconditionally stable and optimally convergent. Several numerical tests are presented for validating the support of the derived theoretical results.