Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-27T13:59:41.177Z Has data issue: false hasContentIssue false
  • English
  • Français

Hermite pseudospectral method for nonlinear partial differential equations

Published online by Cambridge University Press:  15 April 2002

Ben-yu Guo  and
Cheng-long Xu
Ben-yu Guo
Affiliation:
Department of Mathematics, Shanghai Normal University, Shanghai 200234, P.R. China. (byguo@guomai.sh.cn)
Cheng-long Xu
Affiliation:
Department of Mathematics, Shanghai University, Shanghai 201800, P.R. China.
Get access

Abstract

Hermite polynomial interpolation is investigated. Some approximation results are obtained. As an example, the Burgers equation on the whole line is considered. The stability and the convergence of proposed Hermite pseudospectral scheme are proved strictly. Numerical results are presented.

Keywords

Hermite pseudospectral approximationnonlinear partial differential equations.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 4 , July 2000 , pp. 859 - 872
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R.A. Adams, Sobolev Spaces. Academic Press, New York (1975).
C. Bernardi and Y. Maday, Spectral methods, in Techniques of Scientific Computing, Part 2, P.G. Ciarlet and J.L. Lions Eds., Elsevier, Amsterdam (1997) 209-486.
Coulaud, O., Funaro, D. and Kavian, O., Laguerre spectral approximation of elliptic problems in exterior domains. Comp. Mech. Appl. Mech. Eng. 80 (1990) 451-458. CrossRef
Courant, R., Friedrichs, K.O. and Levy, H., Über die partiellen differezengleichungen der mathematischen physik. Math. Annal. 100 (1928) 32-74. CrossRef
D. Funaro, Estimates of Laguerre spectral projectors in Sobolev spaces, in Orthogonal Polynomials and Their Applications, C. Brezinski, L. Gori and A. Ronveaux Eds., Scientific Publishing Co. (1991) 263-266.
Funaro, D. and Kavian, O., Approximation of some diffusion evolution equations in unbounded domains by Hermite functions. Math. Comp. 57 (1990) 597-619. CrossRef
B.Y. Guo, A class of difference schemes of two-dimensional viscous fluid flow. TR. SUST (1965). Also see Acta Math. Sinica 17 (1974) 242-258.
Guo, B.Y., Generalized stability of discretization and its applications to numerical solution of nonlinear differential equations. Contemp. Math. 163 (1994) 33-54. CrossRef
B.Y. Guo, Spectral Methods and Their Applications. World Scientific, Singapore (1998).
Guo, B.Y., Error estimation for Hermite spectral method for nonlinear partial differential equations. Math. Comp. 68 (1999) 1067-1078. CrossRef
A.L. Levin and D.S. Lubinsky, Christoffel functions, orthogonal polynomials, and Nevais conjecture for Freud weights. Constr. Approx. 8 (1992) 461-533.
Lubinsky, D.S. and Moricz, F., The weighted L p -norm of orthogonal polynomial of Freud weights. J. Approx. Theory 77 (1994) 42-50. CrossRef
Y. Maday, B. Pernaud-Thomas and H. Vandeven, Une réhabilitation des méthodes spectrales de type Laguerre. Rech. Aérospat. 6 (1985) 353-379.
R.D. Richitmeyer and K.W. Morton, Finite Difference Methods for Initial Value Problems, 2nd ed., Interscience, New York (1967).
H.J. Stetter, Stability of nonlinear discretization algorithms, in Numerical Solutions of Partial Differential Equations, J. Bramble Ed., Academic Press, New York (1966) 111-123.
G. Szegö, Orthogonal Polynomials. Amer. Math. Soc., New York (1967).
A.F. Timan, Theory of Approximation of Functions of a Real Variable. Pergamon Press, Oxford (1963).