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Journal of Scientific Computing

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01-10-2012

Stability and Convergence Analysis of Fully Discrete Fourier Collocation Spectral Method for 3-D Viscous Burgers’ Equation

Authors: Sigal Gottlieb, Cheng Wang

Published in: Journal of Scientific Computing | Issue 1/2012

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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.

previous article High Order Weighted Essentially Non-oscillation Schemes for Two-Dimensional Detonation Wave Simulations

next article Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering for Discontinuous Galerkin Solutions: Improved Errors Versus Higher-Order Accuracy

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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) MathSciNetMATHCrossRef

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) MATHCrossRef

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) MATHCrossRef

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) MathSciNetMATHCrossRef

Canuto, C., Quarteroni, A.: Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput. 38, 67–86 (1982) MathSciNetMATHCrossRef

Canuto, C., Hussani, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006) MATH

Canuto, C., Hussani, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics. Springer, Berlin (2007) MATH

Chen, G.Q., Du, Q., Tadmor, E.: Super viscosity approximations to multi-dimensional scalar conservation laws. Math. Comput. 61(204), 629–643 (1993) MathSciNetMATHCrossRef

10.

Crouzeix, M.: Une methode multipas implicite-explicite pour l’approximation des’equations d’evolution paraboliques. Numer. Math. 35, 257–276 (1982) MathSciNetCrossRef

11.

De Frutos, J., Ortega, T., Sanz-Serna, J.M.: Pseudo-spectral method for the “Good” Boussinesq equation. Math. Comput. 57(195), 109–122 (1991) MATH

12.

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) MathSciNetMATHCrossRef

13.

Faure, S., Laminie, J., Temam, R.: Finite volume discretization and multilevel methods in flow problems. J. Sci. Comput. 25(112), 231–261 (2005) MathSciNetMATH

14.

Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods, Theory and Applications. SIAM, Philadelphia (1977) MATHCrossRef

15.

Gottlieb, D., Temam, R.: Implementation of the nonlinear Galerkin method with pseudospectral (collocation) discretizations. Appl. Numer. Math. 12, 119–134 (1993) MathSciNetMATHCrossRef

16.

Guo, B.Y.: A spectral method for the vorticity equation on the surface. Math. Comput. 64(211), 1067–1069 (1995) MATH

17.

Guo, B.Y., Huang, W.: Mixed Jacobi-spherical harmonic spectral method for Navier–Stokes equations. Appl. Numer. Math. 57(8), 939–961 (2007) MathSciNetMATHCrossRef

18.

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 MathSciNetMATH

19.

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) MathSciNetMATHCrossRef

20.

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) MathSciNetMATHCrossRef

21.

Guo, B.Y., Ma, H.P., Tadmor, E.: Spectral vanishing viscosity method for nonlinear conservation laws. SIAM J. Numer. Anal. 39, 1254–1268 (2001) MathSciNetMATHCrossRef

22.

Hesthaven, J., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems. Cambridge University Press, Cambridge (2007) MATHCrossRef

23.

Karniadakis, G., Israeli, M., Orszag, S.: High-order splitting methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 97, 414–443 (1991) MathSciNetMATHCrossRef

24.

Kreiss, H.O., Oliger, J.: Methods for Approximate Solution of Time-Dependent Problems. GARP Publications, vol. 10. World Meteorological Organization, Geneva (1973)

25.

Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-Linear Equations of Parabolic Type. AMS, Providence (1967)

26.

Le Quere, P.: Transition to unsteady natural convection in a tall water-filled cavity. Phys. Fluids A 2, 503–515 (1990) MATHCrossRef

27.

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) MathSciNetMATHCrossRef

28.

Ma, H.P.: Chebyshev-Legendre super spectral viscosity method for nonlinear conservation laws. SIAM J. Numer. Anal. 35, 893–908 (1998) MathSciNetMATHCrossRef

29.

Maday, Y., Quarteroni, A.: Legendre and Chebyshev spectral approximations of Burgers’ equation. Numer. Math. 37, 321–332 (1981) MathSciNetMATHCrossRef

30.

Maday, Y., Quarteroni, A.: Approximation of Burgers’ equation by pseudospectral methods. RAIRO. Anal. Numér. 16, 375–404 (1982) MathSciNetMATH

31.

Maday, Y., Quarteroni, A.: Spectral and pseudospectral approximation to Navier-Stokes equations. SIAM J. Numer. Anal. 19(4), 761–780 (1982) MathSciNetMATHCrossRef

32.

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) MathSciNetMATHCrossRef

33.

Majda, A., McDonough, J., Osher, S.: The Fourier method for non-smooth initial data. Math. Comput. 32, 1041–1081 (1978) MathSciNetMATHCrossRef

34.

Marion, M., Temam, R.: Nonlinear Galerkin methods. SIAM J. Numer. Anal. 26, 1139–1157 (1989) MathSciNetMATHCrossRef

35.

Marion, M., Temam, R.: Nonlinear Galerkin methods: the finite element case. Numer. Math. 57, 205–226 (1990) MathSciNetMATHCrossRef

36.

Peyret, R.: Spectral Methods for Incompressible Viscous Flow. Springer, New York (2001)

37.

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) MathSciNetMATHCrossRef

38.

Tadmor, E.: The exponential accuracy of Fourier and Chebyshev differencing methods. SIAM J. Numer. Anal. 23, 1–10 (1986) MathSciNetMATHCrossRef

39.

Tadmor, E.: Convergence of spectral methods to nonlinear conservation laws. SIAM J. Numer. Anal. 26(1), 30–44 (1989) MathSciNetMATHCrossRef

40.

Tadmor, E.: Shock capturing by the spectral viscosity method. Comput. Methods Appl. Mech. Eng. 80, 197–208 (1990) MathSciNetMATHCrossRef

41.

Tadmor, E.: Total variation and error estimates for spectral viscosity approximations. Math. Comput. 60(201), 245–256 (1993) MathSciNetMATHCrossRef

42.

Tadmor, E.: Burgers’ equation with vanishing hyper-viscosity. Commun. Math. Sci. 2(2), 317–324 (2004) MathSciNetMATH

43.

Temam, R.: Navier-Stokes Equations: Theory and Numerical Analysis. AMS, Providence (2001) MATH

44.

Weinan, E.: Convergence of spectral methods for the Burgers’ equation. SIAM J. Numer. Anal. 29(6), 1520–1541 (1992) MathSciNetMATHCrossRef

45.

Weinan, E.: Convergence of Fourier methods for Navier-Stokes equations. SIAM J. Numer. Anal. 30(3), 650–674 (1993) MathSciNetMATHCrossRef

46.

Wu, S., Liu, X.: Convergence of spectral method in time for the Burgers’ equation. Acta Math. Appl. Sin. 13(3), 314–320 (1997) MATHCrossRef

Title: Stability and Convergence Analysis of Fully Discrete Fourier Collocation Spectral Method for 3-D Viscous Burgers’ Equation
Authors: Sigal Gottlieb
Cheng Wang
Publication date: 01-10-2012
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2012
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-012-9621-8

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