Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

On the stability of Galerkin methods for initial-boundary value problems for hyperbolic systems


Author: Max D. Gunzburger
Journal: Math. Comp. 31 (1977), 661-675
MSC: Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-1977-0436624-0
MathSciNet review: 0436624
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The stability of approximating the solution of mixed initial-boundary value problems for hyperbolic systems by semidiscrete Galerkin methods is studied. It is shown that a particular straightforward Galerkin method yields an unstable approximation, and that this numerical instability is caused by an improper treatment of the boundary. Stable schemes are then presented, one of which differs from the unstable scheme only insofar as the treatment of the boundary is concerned. These stable schemes make use of a particular matrix which symmetrizes the differential system. It is therefore shown that the use of this matrix is crucial to the stability of the computations as well as for obtaining a priori bounds on the energy of the continuous system. This symmetrizing matrix is also related to the diagonalizing matrix for the system of hyperbolic equations and to the Lyapunov matrix for the system of ordinary differential equations resulting from the application of Galerkin's method.


References [Enhancements On Off] (What's this?)

  • [1] G. A. BAKER, "A finite element method for first order hyperbolic equations," Math. Comp., v. 29, 1975, pp. 995-1006. MR 0400744 (53:4574)
  • [2] J. E. DENDY, "Two methods of Galerkin type achieving optimum $ {L^2}$ rates of convergence for first order hyperbolics," SIAM J. Numer. Anal., v. 11, 1974, pp. 637-653. MR 50 # 6178. MR 0353695 (50:6178)
  • [3] T. DUPONT, "Galerkin methods for first order hyperbolics: an example," SIAM J. Numer. Anal., v. 10, 1973, pp. 890-899. MR 50 # 1540. MR 0349046 (50:1540)
  • [4] T. DUPONT, "A Galerkin method for liquid pipelines." (To appear.)
  • [5] I. FRIED, "Condition of finite element matrices generated from nonuniform meshes," AIAA J., v. 10, 1972, pp. 219-221.
  • [6] H.-O. KREISS & J. OLIGER, Methods for the Approximate Solution of Time Dependent Problems, Global Atmospheric Research Programme, Publications Series, no. 10, Geneva, 1973.
  • [7] O. TAUSSKY, "Positive-definite matrices and their role in the study of the characteristic roots of general matrices," Advances in Math., v. 2, 1968, pp. 175-186. MR 37 # 2785. MR 0227200 (37:2785)
  • [8] L. WAHLBIN, "A modified Galerkin procedure with Hermite cubics for hyperbolic problems," Math. Comp., v. 29, 1975, pp. 978-984. MR 52 # 9643. MR 0388809 (52:9643)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N30

Retrieve articles in all journals with MSC: 65N30


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1977-0436624-0
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society