Abstract
A Chebyshev-Fourier approximation to the solution of the two-dimensional Stokes equations in the vorticity-streamfunction formulation is considered. The expansion in a Fourier series in the direction of periodicity leads to a family of one-dimensional Stokes-type problems being approximated by a Chebyshevcollocation method. First some results about the spectrum of the corresponding operators are derived. Then we consider a discretization in time by means of a class of semi-implicit finite differences schemes and we describe the influence matrix technique used to solve the resulting system at every time step. The properties of the spectrum of the Chebyshev-Stokes operator are used to derive some results about the stability of the resulting time marching algorithm.
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References
Canuto, C., Hussaini, A., Quarteroni, A., and Zang, T. A. (1987).Spectral Methods in Fluid Dynamics. Springer Series in Computational Physics, Springer, New York.
Canuto, C., and Sacchi Landriani, G. (1986). Analysis of the Kleiser-Schumann method,Numer. Math. 50, 217–243.
Demay, Y., Lacroix, J. M., Peyret, R., and Vanel, J. M. (1987). Numerical experiments on stratified fluid subject to heating, Third International Symposium on Stratified Flows, Pasadena, California, February 3–5.
Ehrenstein, U. (1986). Méthodes spectrales de résolution des équations de Stokes et de Navier-Stokes. Applicationá des écoulements de convection double-diffusive, thesis, Université de Nice.
Ehrenstein, U., and Peyret, R. (1986). A collocation Chebyshev method for solving Stokes-type equations,Sixth Int. Symp. Finite Element Methods in Flows Problems, Antibes, June 16–20.
Ehrenstein, U., and Peyret, R. (1989). A Chebyshev-collocation method for the Navier-Stokes equations with application to double-diffusive convection,Int. J. Numer. Methods Fluids,9, 499–515.
Gantmacher, F. R. (1960).Matrix Theory, Vol. II, Chelsea Publishing, New York.
Gottlieb, D., Hussaini, M. Y., and Orszag, S. A. (1984). Theory and application of spectral methods, inSpectral Methods for Partial Differential Equations, Voigt, R. G., Gottlieb, D., and Hussaini, M. Y. (eds.), pp. 1–54, SIAM, Philadelphia.
Gottlieb, D., and Lustman, L. (1983). The spectrum of the Chebyshev collocation operator for the heat equation,SIAM J. Numer. Anal. 20, (5), 909–921.
Gottlieb, D., and Orszag, S. A. (1977).Numerical Analysis of Spectral Methods: Theory and Applications, BMS Regional Conference Series in Appl. Math., SIAM, Philadelphia.
Gourlay, A. R., and Griffiths, D. F. (1980).The Finite Difference Method in Partial Differential Equations, John Wiley and Sons, New York.
Kleiser, L., and Schumann, U. (1980). Treatment of incompressibility and boundary conditions in 3-D numerical spectral simulations of plane channel flows,Proc. Third GAMM-Conference on Numerical Methods in Fluid Mechanics, Hirschel, E. H. (ed.), pp. 165–173, Vieweg, Braunschweig.
Miller, J. J. H. (1971). On the location of zeros of certain classes of polynomials with application to numerical analysis,J. Int. Math. Appl. 8, 397–406.
Ouazzani, J., Peyret, R., and Zakaria, A. (1986). Stability of collocation-Chebyshev schemes with application to the Navier-Stokes equations,Proc. Sixth GAMM Conf. on Numer. Methods in Fluid Mechanics, Rues, D., and Kordulla, W. (eds.), pp. 287–294, Vieweg, Braunschweig.
Peyret, R. (1988).Introduction to Spectral Methods, von Karman Institute Lecture Series 1986, 4, Rhode, Saint Genese, Belgium.
Pulicani, J.-P. (1988). Application des méthodes spectrales à l'étude d'écoulements de convection, thesis, Université de Nice.
Le Quere, P., and Alziary de Roquefort, T. (1985). Computation of natural convection in twodimensional cavities with Chebyshev polynomials,J. Comput. Phys. 57, 210–228.
Tuckermann, L. (1983). Formation Taylor vortices in spherical Couette flow, Ph. D. thesis, M.I.T., Cambridge.
Vanel, J. M., Peyret, R., and Bontoux, P. (1986). A pseudo-spectral solution of vorticity-stream function equations using the influence matrix technique, inNumerical Methods for Fluid Dynamics II, Morton, K. W., and Baines, M. J. (eds.), pp. 463–475, Clarendon Press, Oxford.
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Ehrenstein, U. The spectrum of a Chebyshev-Fourier approximation for the Stokes equations. J Sci Comput 5, 55–84 (1990). https://doi.org/10.1007/BF01063426
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DOI: https://doi.org/10.1007/BF01063426