Long time unconditional stability of a two-level hybrid method for nonstationary incompressible Navier–Stokes equations

Abstract

This paper presents a two-level MacCormack rapid solver method for solving a 2D time dependent Navier–Stokes problem that models an incompressible fluid flow. A decoupling approach based on the coupling term via spatial extrapolation is proposed for devising decoupled marching algorithms for the nonstationary incompressible model. In this hybrid method, an explicit MacCormack scheme provides a H 1 -optimal velocity approximation and a L 2 -optimal pressure approximation by solving a nonlinear Navier–Stokes problem on a coarse grid with mesh size H , while an implicit Crank–Nicolson algorithm consists in dealing with the fully discrete linear generalized Stokes problem on a fine grid with mesh size h ≪ H . The theoretical result suggests that our method is unconditionally stable over long time intervals. While the numerical evidences both confirm the theoretic analysis and show that the algorithm is convergent, the tests also suggest that our method is both cheaper and faster than the two-level finite element Galerkin scheme.