Abstract
A pseudo-spectral (or collocation) approximation of the unsteady Stokes equations is presented. Using the Uzawa algorithm the spectral system is decoupled into Helmholtz equations for the velocity components and an equation with the Pseudo-Laplacian for the pressure. In order to avoid spurious modes the pressure is approximated with lower order (two degrees lower) polynomials than the velocity. Only one grid (no staggered grids) with the standard Chebyshev Gauss-Lobatto nodes is used. Here we further compare our treatment with a Neumann boundary value problem for the pressure. The highly improved accuracy of our method becomes obvious. In the time discretization a high order backward differentiation scheme for the intermediate velocity is combined with a high order extrapolant for the pressure. It is numerically shown that a stable third order method in time can be achieved.
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References
Arrow, K., Hurwicz, L., and Uzawa, H. (1968).Studies in Nonlinear Programming. Stanford University Press, Stanford.
Azaiez, M. (1990). Calcul de la pression dans le problème de Stokes par une méthode pseudospectrale de collocation, Thèse, Université de Paris Sud.
Azaiez, M., Fikoi, A., and Labrosse, G. (1994). A unique grid spectral solver of the not cartesian unsteady Stokes system. Illustrative numerical results.Finite Elements in Analysis and Design 16, 247–260.
Bernardi, C., and Maday, Y. (1988). A collocation method over staggered grids for the Stokes problem.Int. J. Num. Meth. Fluids 8, 537–557.
Brandt, A., Fulton, S. R., and Taylor, G. D. (1985). Improved spectral multigrid methods for periodic elliptic problems.J. Comp. Phys. 58, 96–112.
Bristeau, M. O., Glowinski, R., and Periaux, J. (1987). Numerical methods for the Navier-Stokes equations. Applications to the simulation of compressible and incompressible viscous flows.Computer Physics Report 6, 73–187.
Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A. (1989). Spectral methods in fluid dynamics,Springer Series in Computational Physics, Springer-Verlag, Berlin-Heidelberg-New York.
Girault, V., and Raviart, P. (1988).Finite Element Approximation of the Navier-Stokes Equations. Springer-Verlag.
Glowinski R. (1984).Numerical Methods for Nonlinear Variational Problems. Springer-Verlag.
Glowinski, R., and Pironneau, O. (1979). On a mixed finite element approximation of the Stokes problem.Numer. Math. 33, 397–424.
Gottlieb, D., and Orszag, S. A. (1977). Numerical analysis of spectral methods: theory and applications. CBMS-NSF Regional Conference Series in Applied Mathematics No. 26, SIAM, Philadelphia.
Heinrichs, W. (1987). Kollokationsverfahren und Mehrgittermethoden bei elliptischen Randwertaufgaben. GMD-Bericht Nr. 168, Oldenbourg-Verlag, München-Wien.
Heinrichs, W. (1988a). Line relaxation for spectral multigrid methods.J. Comp. Phys. 77, 166–182.
Heinrichs, W. (1988b). Multigrid methods for combined finite difference and Fourier problems.J. Comp. Phys. 78, 424–436.
Heinrichs, W. (1990). Algebraic spectral multigrid methods. InSpectral and High Order Methods for Partial Differential Equations. Proc. of the ICOSAHOM'89 Conf., Canuto, C., and Quarteroni, A. (eds.), Como, Elsevier Sci. Publishers B. V., North-Holland, andComp. Meth. Appl. Mech. Eng. 80, 281–286.
Heinrichs, W. (1991). A 3D spectral multigrid method.Appl. Math Comp. 41, 117–128.
Heinrichs, W. (1993a). Distributive relaxations for the spectral Stokes operator.J. Sci. Comp. 8, (4) 389–398.
Heinrichs, W. (1993b). Spectral multigrid techniques for the Navier-Stokes equations.Comput. Meth. Appl. Mech. Eng. 106, 297–314.
Heinrichs, W. (1993c). Splitting techniques for the pseudospectral approximation of the unsteady Stokes equations.SIAM J. Numer. Anal. 30, 19–39.
Kleiser, L., and Schumann, U. (1984). Spectral simulation of the laminar turbulent transition process in plane Poiseuille flow. InSpectral Methods for Partial Differential Equations, Voigt, R. G., Gottlieb D., and Hussaini, M. Y. (eds.), SIAM.
Maday, Y., Patera, A. T., and Rønquist, E. (1990). An operator-integration factor splitting method for time-dependent problems: application to incompressible fluid flow.J. Sci. Comput. 5, 263–292.
Orszag, S. A. (1980). Spectral methods for problems in complex geometries.J. Comp. Phys. 37, 70–92.
Quazzani, J., Peyret, R., and Zakaria, A. (1986). Stability of collocation-Chebyshev schemes with applications to the Navier-Stokes equations, InProc. Sixth GAMM Conf. on Numerical Methods in Fluid Mechanics, Rues D., and Kordulla, W. (eds.), Braunschweig.
Rannacher, R. (1991). On Chorin's projection method for the incompressible Navier-Stokes equations, InThe Navier-Stokes Equations II—Theory and Numerical Methods, Heywood, J. G., Masuda, K., Rautmann R., and Solonnikov, S. A. (eds.),Lecture Notes in Mathematics, Springer-Verlag, Berlin-Heidelberg-New York.
Rønquist, E. (1988). Optimal spectral element methods for the unsteady three-dimensional incompressible Navier-Stokes equations. Ph. D. Thesis, M.I.T, Cambridge.
Schumack, M. R., Schultz, W. M., and Boyd, J. P. (1991). Spectral method solution of the Stokes equations on nonstaggered grids.J. Comp. Phys. 94, 30–58.
Temam, R. (1984).Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam.
Yamaguchi, Y., Chang, C. J., and Brown, R. A. (1984). Multiple bouyancy-driven flows in a vertical cylinder heated from below.Phil. Trans. Roy. Soc. London A 312, 519.
Zang, T. A., Wong, Y. S., and Hussaini, M. Y. S. (1982). Spectral multigrid methods for elliptic equations I.J. Comp. Phys. 48, 485–501.
Zang, T. A., Wong, Y. S., and Hussaini, M. Y. (1984). Spectral multigrid methods for elliptic equations II.J. Comp. Phys. 54, 489–507.
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Heinrichs, W. High-order time splitting for the Stokes equations. J Sci Comput 11, 397–410 (1996). https://doi.org/10.1007/BF02088954
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DOI: https://doi.org/10.1007/BF02088954