A collocation-Galerkin method for a first order hyperbolic equation with space and time dependent coefficient

Authors:
David Archer and Julio César Díaz

Journal:
Math. Comp. **38** (1982), 37-53

MSC:
Primary 65M60; Secondary 65N35

DOI:
https://doi.org/10.1090/S0025-5718-1982-0637285-X

MathSciNet review:
637285

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Abstract: A collocation-Galerkin scheme is proposed for an initial-boundary value problem for a first order hyperbolic equation in one space dimension. The Galerkin equations satisfied by the approximating solution are obtained from a weak-weak formulation of the initial-boundary value problem. The collocation points are taken to be affine images of the roots of the Jacobian polynomials of degree on [0, 1] with respect to the weight function . Optimal -norm estimates of the error are derived.

**[1]**D. Archer, ``Collocation in spaces for first order hyperbolic equations. I: Optimal order global estimates for quasilinear problems,''*SIAM J. Numer. Anal.*, v. 15, 1978, pp. 271-281. MR**0474889 (57:14518)****[2]**G. A. Baker, ``A finite element method for first order hyperbolic equations,''*Math Comp.*, v. 29, 1975, pp. 995-1006. MR**0400744 (53:4574)****[3]**J. C. Diaz Velasco, Ph. D. Thesis, Rice University, Houston, Texas, 1974.**[4]**J. C. Diaz, ``A collocation-Galerkin method for the two point boundary value problem using continuous piecewise polynomial spaces,''*SIAM J. Numer. Anal.*, v. 14, 1977, pp. 844-858. MR**0483480 (58:3481)****[5]**J. C. Diaz, ``Collocation -Galerkin method for parabolic problems with time dependent coefficients,''*SIAM J. Numer. Anal.*, v. 16, 1979, pp. 911-922. MR**551315 (81b:65099)****[6]**R. J. Dunn, Jr. & M. F. Wheeler, ``Some collocation-Galerkin methods for two-point boundary value problems,''*SIAM J. Numer. Anal.*, v. 13, 1976, pp. 720-723. MR**0433896 (55:6867)****[7]**M. F. Wheeler, ``An elliptic collocation-finite element method with interior-penalties,''*SIAM J. Numer. Anal.*, v. 15, 1978, pp. 152-161. MR**0471383 (57:11117)**

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DOI:
https://doi.org/10.1090/S0025-5718-1982-0637285-X

Article copyright:
© Copyright 1982
American Mathematical Society