Dispersion and Gibbs phenomenon associated with difference approximations to initial boundary-value problems for hyperbolic equations

https://doi.org/10.1016/0021-9991(75)90001-7Get rights and content

Abstract

In this paper, we analyze the problem of the semidiscretized approximation for the initial boundary-value problem of the wave equation. Point-wise convergence properties for the propagation of discontinuities are investigated via a uniformly valid asymptotic expansion. An approximate error analysis using matched asymptotic expansions is constructed and compared with the asymptotic expansion of the exact solution.

References (17)

  • S.I. Serdyukova

    USSR Comput. Math. Math. Phys.

    (1971)
  • S.A. Orszag et al.

    J. Computational Phys.

    (1974)
  • R.C.Y. Chin

    J. Computational Phys.

    (1974)
  • G.W. Hedstrom

    SIAM J. Numer. Anal.

    (1968)
  • A.I. Zhukov

    Usp. Mat. Nauk.

    (1959)
  • C.E. Pearson

    Math. Comp.

    (1969)
  • V. Thomee

    On the Rate of Convergence of Difference Schemes for Hyperbolic Equations

  • M.Y.T. Apelkrans

    Math. Comp.

    (1968)
There are more references available in the full text version of this article.

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This work was performed under the auspices of the U.S. Atomic Energy Commission.

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