Algebraic fractional-step schemes with spectral methods for the incompressible Navier–Stokes equations

The numerical investigation of a recent family of algebraic fractional-step methods for the solution of the incompressible time-dependent Navier–Stokes equations is presented. These methods are improved versions of the Yosida method proposed in [A. Quarteroni, F. Saleri, A. Veneziani, Factorization methods for the numerical approximation of Navier–Stokes equations Comput. Methods Appl. Mech. Engrg. 188(1–3) (2000) 505–526; A. Quarteroni, F. Saleri, A. Veneziani, J. Math. Pures Appl. (9), 78(5) (1999) 473–503] and one of them (the Yosida4 method) is proposed in this paper for the first time. They rely on an approximate LU block factorization of the matrix obtained after the discretization in time and space of the Navier–Stokes system, yielding a splitting in the velocity and pressure computation. In this paper, we analyze the numerical performances of these schemes when the space discretization is carried out with a spectral element method, with the aim of investigating the impact of the splitting on the global accuracy of the computation.