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Error estimation of Hermite spectral method
for nonlinear partial differential equations

Author: Ben-yu Guo
Journal: Math. Comp. 68 (1999), 1067-1078
Published electronically: February 5, 1999
MathSciNet review: 1627789
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Abstract | References | Additional Information

Abstract: Hermite approximation is investigated. Some inverse inequalities, imbedding inequalities and approximation results are obtained. A Hermite spectral scheme is constructed for Burgers equation. The stability and convergence of the proposed scheme are proved strictly. The techniques used in this paper are also applicable to other nonlinear problems in unbounded domains.

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  • 1. Y. Maday, B. Pernaud-Thomas and H. Vandeven, One réhabilitation des méthodes spectrales de type Laguerre, Rech. Aérospat., 6 (1985), 353-379.
  • 2. O. Coulaud, D. Funaro and O. Kavian, Laguerre spectral approximation of elliptic problems in exterior domains, Comput. Methods Appl. Mech. Engrg., 80 (1990), 451-458. CMP 90:17
  • 3. D. Funaro, Estimates of Laguerre spectral projectors in Sobolev spaces, in Orthogonal Polynomials and Their Applications, ed. by C. Brezinski, L. Gori and A. Ronveaux, Scientific Publishing Co., 1991, 263-266. MR 95a:41033
  • 4. V. Iranzo and A. Falquès, Some spectral approximations for differential equations in unbounded domains, Comput. Methods Appl. Mech. Engrg., 98 (1992), 105-126. MR 93d:65103
  • 5. C. Mavriplis, Laguerre polynomials for infinite-domain spectral elements, J. Comp. Phys., 80 (1989), 480-488. MR 90f:65228
  • 6. K. Black, Spectral elements on infinite domains (unpublished).
  • 7. D. Funaro and O. Kavian, Approximation of some diffusion evolution equations in unbounded domains by Hermite funtions, Math. Comp., 57 (1990), 597-619. MR 92k:35156
  • 8. J. A. C. Weideman, The eigenvalues of Hermite and rational spectral differential matrices, Numer. Math., 61 (1992), 409-432. MR 92k:65071
  • 9. C. I. Christov, A complete orthonormal system of funtions in $L^2(- \infty, \infty)$ space, SIAM J. Appl. Math., 42 (1982), 1337-1344. MR 84b:42018
  • 10. J. P. Boyd, Spectral methods using rational basis funtions on an infinite interval, J. Comp. Phys., 69 (1987), 112-142. MR 88e:65093
  • 11. R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. MR 56:9247
  • 12. R. J. Nessel and G. Wilmes, On Nikolskii-type inequalities for orthogonal expansion, in Approximation Theory II , ed. by G. G. Lorentz, C. K. Chui and L. L. Schumaker, Academic Press, New York, 479-484.
  • 13. R. Courant, K. O. Friedrichs and H. Lewy, $\ddot \mathrm{ U}$ber die partiellen differezengleichungen der mathematischen physik, Math. Ann., 100 (1928), 32-74.
  • 14. B. Y. Guo, A certain class of finite difference schemes of two-dimensional vorticity equation viscous fluid, RR. SUST, 1965, Also see Acta Math. Sinica, 17 (1974), 242-258. MR 56:17128
  • 15. B. Y. Guo, Generalized stability of discretization and its applications to numerical solutions of nonlinear differential equations, Contemp. Math., 163 (1994), 33-54. MR 95d:65067
  • 16. H. J. Stetter, Stability of nonlinear discretization algorithms, in Numerical Solutions of Partial Differential Equations, ed. by J. Bramble, Academic Press, New York, 1966, 111-123. MR 34:5322
  • 17. G. H. Hardy, J. E. Littlewood and G.Pólya, Inequalities, 2'ed., Cambridge University Press, Cambridge, 1952. MR 13:727e

Additional Information

Ben-yu Guo
Affiliation: Department of Mathematics, Shanghai University, Jiading Campus, Shanghai, 201800, China
Address at time of publication: Department of Mathematics, Shanghai Normal University, Shanghai, China

Keywords: Hermite approximation, Burgers equation, error estimations
Received by editor(s): October 16, 1997
Received by editor(s) in revised form: January 2, 1998
Published electronically: February 5, 1999
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society