# zbMATH — the first resource for mathematics

On the solution of the Navier-Stokes equations using Chebyshev projection schemes with third-order accuracy in time. (English) Zbl 0898.76077
Summary: A third-order time-accurate projection method for approximating the Navier-Stokes equations for incompressible flow is presented. In order to compute a pressure unpolluted by spurious modes, two Chebyshev collocation spatial discretizations, where the pressure is approximated by lower-order polynomials than the velocity, are compared. One only collocation grid is used, and no pressure boundary condition is needed. The Navier-Stokes problem is reduced to the successive solution of Helmholtz problems for the velocity and pseudo-Poisson problems for the pressure. These problems are solved by direct methods. Using an exact solution, spectral spatial accuracy, and third-order time accuracy, for both the velocity and the pressure, are checked. The stability properties are discussed by considering the regularized cavity flow at different Reynolds numbers.

##### MSC:
 76M25 Other numerical methods (fluid mechanics) (MSC2010) 76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text:
##### References:
 [1] Chorin, A., Numerical simulation of the Navier-Stokes equations, Math. comput., 22, 745-762, (1968) · Zbl 0198.50103 [2] Temam, R., Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires II, Archiv. rat. mech. anal., 32, 377-385, (1969) · Zbl 0207.16904 [3] Temam, R., () [4] Temam, R., Remark on the pressure boundary condition for the projection method, Theoret. comput. fluid dynamics, 3, 181-184, (1991) · Zbl 0738.76054 [5] Guermond, J.-L., Remarques sur LES méthodes de projection pour l’approximation des équations de Navier-Stokes, Numer. math., 67, 465-473, (1994) · Zbl 0802.76057 [6] Perot, J.B., An analysis of the fractional step method, J. computat. phys., 108, 51-58, (1992) · Zbl 0778.76064 [7] Azaïez, M.; Bernardi, C.; Grundmann, M., Spectral methods applied to porous media equations, East-west J. numer. math., 2, 91-105, (1994) · Zbl 0838.76055 [8] Morchoisne, Y., Résolution des équations de Navier-Stokes par une méthode spectrale de sous-domaines, (), 181-208, Paris [9] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., () [10] Maday, Y.; Patera, A.T.; Rønquist, E.M., The P_{N} × PN − 2 method for the approximation of the Stokes problem, Laboratoire d’analyse numérique, Paris VI, 11, 4, (1992) [11] Heinrichs, W., Splitting techniques for the pseudospectral approximation of the unsteady Stokes equations, SIAM J. numer. anal., 30, 19-39, (1993) · Zbl 0764.76054 [12] Heinrichs, W., Operator splitting for the Stokes equations, to appear. · Zbl 0911.76058 [13] Azaïez, M.; Fikri, A.; Labrosse, G., A unique grid spectral solver of the nd Cartesian unsteady Stokes system. illustrative numerical results, Finite elements in analysis and design, 16, 247-260, (1994) · Zbl 0812.76057 [14] Quarteroni, A.; Valli, A., () [15] Peyret, R.; Taylor, T.D., () [16] Heywood, J.C.; Rannacher, R., Finite element approximation of the nonstationary Navier-Stokes problem. I: regularity of the solutions and second-order error estimates for spatial discretization, SIAM J. num. anal., 19, 275-311, (1982) · Zbl 0487.76035 [17] Ehrenstein, U.; Peyret, R., A Chebyshev-collocation method for the Navier-Stokes equations with application to double-diffusive convection, Int. J. numer. methods fluids, 9, 427-452, (1989) · Zbl 0665.76107 [18] Botella, O., Résolution des équations de Navier-Stokes par des schémas de projection Tchebychev. INRIA research report (to appear). [19] Shen, J., Hopf bifurcation of the unsteady regularized driven cavity flow, J. computat. phys., 95, 228-245, (1991) · Zbl 0725.76059 [20] Batoul, A.; Khallouf, H.; Labrosse, G., Une méthode de résolution directe (pseudo-spectrale) du problème de Stokes $$2D3D$$ instationnaire. application à la cavité entraî carrée, C. R. acad. sci. Paris, 319, II, 1455-1461, (1994) · Zbl 0816.76062 [21] Goodrich, J.W.; Gustafson, K.; Halasi, K., Hopf bifurcation in the driven cavity, J. computat. phys., 90, 219-261, (1990) · Zbl 0702.76052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.