Abstract
This paper analyzes the stability and convergence of the Fourier pseudospectral method coupled with a variety of specially designed time-stepping methods of up to fourth order, for the numerical solution of a three dimensional viscous Burgers’ equation. There are three main features to this work. The first is a lemma which provides for an L 2 and H 1 bound on a nonlinear term of polynomial type, despite the presence of aliasing error. The second feature of this work is the development of stable time-stepping methods of up to fourth order for use with pseudospectral approximations of the three dimensional viscous Burgers’ equation. Finally, the main result in this work is that the pseudospectral method coupled with the carefully designed time-discretizations is stable provided only that the time-step and spatial grid-size are bounded by two constants over a finite time. It is notable that this stability condition does not impose a restriction on the time-step that is dependent on the spatial grid size, a fact that is especially useful for three dimensional simulations.
Similar content being viewed by others
References
Abia, L., Sanz-Serna, J.M.: The spectral accuracy of a fully-discrete scheme for a nonlinear third order equation. Computing 44, 187–196 (1990)
Botella, O.: On the solution of the Navier-Stokes equations using projection schemes with third-order accuracy in time. Comput. Fluids 26, 107–116 (1997)
Botella, O., Peyret, R.: Computing singular solutions of the Navier-Stokes equations with the Chebyshev-collocation method. Int. J. Numer. Methods Fluids 36, 125–163 (2001)
Boyd, J.: Chebyshev and Fourier Spectrum Methods, 2nd edn. Dover, New York (2001)
Bressan, A., Quarteroni, A.: An implicit/explicit spectral method for Burgers’ equation. Calcolo 23(3), 265–284 (1986)
Canuto, C., Quarteroni, A.: Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput. 38, 67–86 (1982)
Canuto, C., Hussani, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)
Canuto, C., Hussani, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics. Springer, Berlin (2007)
Chen, G.Q., Du, Q., Tadmor, E.: Super viscosity approximations to multi-dimensional scalar conservation laws. Math. Comput. 61(204), 629–643 (1993)
Crouzeix, M.: Une methode multipas implicite-explicite pour l’approximation des’equations d’evolution paraboliques. Numer. Math. 35, 257–276 (1982)
De Frutos, J., Ortega, T., Sanz-Serna, J.M.: Pseudo-spectral method for the “Good” Boussinesq equation. Math. Comput. 57(195), 109–122 (1991)
Du, Q., Guo, B., Shen, J.: Fourier spectral approximation to a dissipative system modeling the flow of liquid crystals. SIAM J. Numer. Anal. 39(3), 735–762 (2001)
Faure, S., Laminie, J., Temam, R.: Finite volume discretization and multilevel methods in flow problems. J. Sci. Comput. 25(112), 231–261 (2005)
Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods, Theory and Applications. SIAM, Philadelphia (1977)
Gottlieb, D., Temam, R.: Implementation of the nonlinear Galerkin method with pseudospectral (collocation) discretizations. Appl. Numer. Math. 12, 119–134 (1993)
Guo, B.Y.: A spectral method for the vorticity equation on the surface. Math. Comput. 64(211), 1067–1069 (1995)
Guo, B.Y., Huang, W.: Mixed Jacobi-spherical harmonic spectral method for Navier–Stokes equations. Appl. Numer. Math. 57(8), 939–961 (2007)
Guo, B.Y., Shen, J.: On spectral approximations using modified Legendre rational functions: application to the Korteweg-de Vries equation on the half line. Indiana Univ. Math. J. 50, 181–204 (2001). Special issue: Dedicated to Professors Ciprian Foias and Roger Temam
Guo, B.Y., Zou, J.: Fourier spectral projection method and nonlinear convergence analysis for Navier-Stokes equations. J. Math. Anal. Appl. 282(2), 766–791 (2003)
Guo, B.Y., Li, J., Ma, H.P.: Fourier-Chebyshev spectral method for solving three-dimensional vorticity equation. Acta Math. Appl. Sin. 11(1), 94–109 (1995)
Guo, B.Y., Ma, H.P., Tadmor, E.: Spectral vanishing viscosity method for nonlinear conservation laws. SIAM J. Numer. Anal. 39, 1254–1268 (2001)
Hesthaven, J., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems. Cambridge University Press, Cambridge (2007)
Karniadakis, G., Israeli, M., Orszag, S.: High-order splitting methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 97, 414–443 (1991)
Kreiss, H.O., Oliger, J.: Methods for Approximate Solution of Time-Dependent Problems. GARP Publications, vol. 10. World Meteorological Organization, Geneva (1973)
Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-Linear Equations of Parabolic Type. AMS, Providence (1967)
Le Quere, P.: Transition to unsteady natural convection in a tall water-filled cavity. Phys. Fluids A 2, 503–515 (1990)
Liu, C., Shen, J.: A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Physica D 179, 211–228 (2003)
Ma, H.P.: Chebyshev-Legendre super spectral viscosity method for nonlinear conservation laws. SIAM J. Numer. Anal. 35, 893–908 (1998)
Maday, Y., Quarteroni, A.: Legendre and Chebyshev spectral approximations of Burgers’ equation. Numer. Math. 37, 321–332 (1981)
Maday, Y., Quarteroni, A.: Approximation of Burgers’ equation by pseudospectral methods. RAIRO. Anal. Numér. 16, 375–404 (1982)
Maday, Y., Quarteroni, A.: Spectral and pseudospectral approximation to Navier-Stokes equations. SIAM J. Numer. Anal. 19(4), 761–780 (1982)
Maday, Y., Ould Kaber, S.M., Tadmor, E.: Legendre pseudospectral viscosity method for nonlinear conservation laws. SIAM J. Numer. Anal. 30(2), 321–342 (1993)
Majda, A., McDonough, J., Osher, S.: The Fourier method for non-smooth initial data. Math. Comput. 32, 1041–1081 (1978)
Marion, M., Temam, R.: Nonlinear Galerkin methods. SIAM J. Numer. Anal. 26, 1139–1157 (1989)
Marion, M., Temam, R.: Nonlinear Galerkin methods: the finite element case. Numer. Math. 57, 205–226 (1990)
Peyret, R.: Spectral Methods for Incompressible Viscous Flow. Springer, New York (2001)
Shen, J., Temam, R.: Nonlinear Galerkin method using Chebyshev and Legendre polynomials I. The one-dimensional case. SIAM J. Numer. Anal. 32, 215–234 (1995)
Tadmor, E.: The exponential accuracy of Fourier and Chebyshev differencing methods. SIAM J. Numer. Anal. 23, 1–10 (1986)
Tadmor, E.: Convergence of spectral methods to nonlinear conservation laws. SIAM J. Numer. Anal. 26(1), 30–44 (1989)
Tadmor, E.: Shock capturing by the spectral viscosity method. Comput. Methods Appl. Mech. Eng. 80, 197–208 (1990)
Tadmor, E.: Total variation and error estimates for spectral viscosity approximations. Math. Comput. 60(201), 245–256 (1993)
Tadmor, E.: Burgers’ equation with vanishing hyper-viscosity. Commun. Math. Sci. 2(2), 317–324 (2004)
Temam, R.: Navier-Stokes Equations: Theory and Numerical Analysis. AMS, Providence (2001)
Weinan, E.: Convergence of spectral methods for the Burgers’ equation. SIAM J. Numer. Anal. 29(6), 1520–1541 (1992)
Weinan, E.: Convergence of Fourier methods for Navier-Stokes equations. SIAM J. Numer. Anal. 30(3), 650–674 (1993)
Wu, S., Liu, X.: Convergence of spectral method in time for the Burgers’ equation. Acta Math. Appl. Sin. 13(3), 314–320 (1997)
Author information
Authors and Affiliations
Corresponding author
Additional information
S. Gottlieb. This work was supported by grant number FA-9550-09-0208 from the Air Force Office of Scientific Research.
C. Wang. This work was supported by NSF with grant number DMS-1115420.
Rights and permissions
About this article
Cite this article
Gottlieb, S., Wang, C. Stability and Convergence Analysis of Fully Discrete Fourier Collocation Spectral Method for 3-D Viscous Burgers’ Equation. J Sci Comput 53, 102–128 (2012). https://doi.org/10.1007/s10915-012-9621-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-012-9621-8