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Splitting techniques for the Navier-Stokes equations. (English) Zbl 1011.76063

Neittaanmäki, Pekka (ed.) et al., ENUMATH 99. Numerical mathematics and advanced applications. Proceedings of the 3rd European conference, Jyväskylä, Finland, July 26-30, 1999. Singapore: World Scientific. 526-533 (2000).
Summary: We present a pseudo-spectral approximation for Navier-Stokes equations. First, the unsteady Stokes equations are considered. Using Uzawa algorithm, the spectral system is decoupled into Helmholtz equations for velocity components, and an equation with pseudo-Laplacian for the pressure. In order to avoid spurious modes, the pressure is approximated by polynomials of one degree less than those used for the velocity. Two grids are used by taking Gauss nodes for the pressure and standard Chebyshev-Gauss-Lobatto nodes for the velocity. We prove that the eigenvalues of spectral pseudo-Laplacian are real and negative (except one eigenvalue which is zero and belongs to constant mode). Since the spectral pseudo-Laplacian is very ill-conditioned, we present a suitable finite difference preconditioner for an effective iterative solver. In the time discretization, a high-order backward differentiation scheme for intermediate velocity is combined with a high-order extrapolant for the pressure. It is shown numerically that a stable second-order method in time can be achieved.
For the entire collection see [Zbl 0954.00041].

MSC:

76M22 Spectral methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
76M20 Finite difference methods applied to problems in fluid mechanics
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