Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Estimates away from a discontinuity for dissipative Galerkin methods for hyperbolic equations


Author: William J. Layton
Journal: Math. Comp. 36 (1981), 87-92
MSC: Primary 65N30; Secondary 65M15
DOI: https://doi.org/10.1090/S0025-5718-1981-0595043-8
MathSciNet review: 595043
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the approximate solution of the initial value problem

$\displaystyle \frac{{\partial u}}{{\partial t}} = \frac{{\partial u}}{{\partial x}},\quad u(x,0) = v(x),$

by a dissipative Galerkin method. When v is taken to have a jump discontinuity at zero, that discontinuity will propagate along $ x + t = 0$, in the true solution u. Estimates in $ {L_2}$ and $ {L_\infty }$ of the pollution effects of the discontinuity are found. These estimates show those effects to decay exponentially in $ {h^{ - 1}}$ in regions a fixed distance d from the discontinuity and exponentially in d for fixed h.

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

  • [1] M. Y. T. Apelkrans, "On difference schemes for hyperbolic equations with discontinuous initial values," Math. Comp., v. 22, 1968, pp. 525-539. MR 0233527 (38:1848)
  • [2] Ph. Brenner & V. Thomée, "Stability and convergence rates in $ {L_p}$, for certain difference schemes," Math. Scand., v. 27, 1970, pp. 5-23. MR 0278549 (43:4279)
  • [3] Ph. Brenner & V. ThomÉe, "Estimates near discontinuities for some difference schemes," Math. Scand., v. 28, 1971, pp. 329-340. MR 0305613 (46:4743)
  • [4] A. Calderón, F. Spitzer & H. Widom, "Inversion of Toeplitz matrices," Illinois J. Math., v. 3, 1959, pp. 490-498. MR 0121652 (22:12386)
  • [5] 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 0353695 (50:6178)
  • [6] G. W. Hedstrom, "The rate of convergence of some difference schemes," SIAM J. Numer. Anal., v. 5, 1968, pp. 363-406. MR 0230489 (37:6051)
  • [7] W. J. Layton, Ph. D. Thesis, University of Tennessee, 1980.
  • [8] R. Richards, Uniform Spline Interpolation Operators in $ {L_2}$, MRC Tech. Report # 1305, University of Wisconsin, November 1972.
  • [9] R. D. Richtmyer & K. W. Morton, Difference Methods for Initial Value Problems, 2nd ed., Interscience, New York, 1967. MR 0220455 (36:3515)
  • [10] I. J. Schoenberg, "Cardinal interpolation and spline functions. II," J. Approx. Theory, v. 6, 1972, pp. 404-420. MR 0340899 (49:5649)
  • [11] I. J. Schoenberg, Cardinal Spline Interpolation, Regional Conference Series in Applied Math., # 12, SIAM, Philadelphia, Pa., 1973. MR 0420078 (54:8095)
  • [12] S. I. Serdyukova, "Oscillations which occur in the numerical computation of the discontinuous solutions of differential equations," Ž. Vyčisl. Mat. i Mat. Fiz.,v. 11, 1971, pp. 411-424.
  • [13] V. Thomée, "Spline approximation and difference schemes for the heat equation," The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A. K. Aziz, Ed.), Academic Press, New York, 1972. MR 0403265 (53:7077)
  • [14] V. Thomée, "Convergence estimates for semidiscrete Galerkin methods for initial value problems," Numerische, insbesondere approximationstheoretische Behandlung von Funktionalgleichungen, Lecture Notes in Math.,v. 333, Springer, Berlin, 1973. MR 0458948 (56:17147)
  • [15] V. Thomée & B. Wendroff, "Convergence estimates for Galerkin methods for variable coefficient initial value problems," SIAM J. Numer. Anal., v. 11, 1974, pp. 1059-1068. MR 0371088 (51:7309)
  • [16] R. S. Varga, Functional Analysis and Approximation Theory in Numerical Analysis, Regional Convergence Series in Applied Math., #3, SIAM, Philadelphia, Pa., 1971. MR 0310504 (46:9602)
  • [17] L. B. Wahlbin, "A dissipative Galerkin method for the numerical solution of first order hyperbolic equations," Mathematical Aspects of Finite Elements in Partial Differential Equations (C. de Boor, Ed.), Academic Press, New York, 1974. MR 0658322 (58:31929)
  • [18] L. B. Wahlbin, "A dissipative Galerkin method applied to some quasilinear hyperbolic equations," R.A.I.R.O., v. 8, 1974, pp. 109-117. MR 0368447 (51:4688)
  • [19] L. B. Wahlbin, "A modified Galerkin procedure with Hermite cubics for hyperbolic problems," Math. Comp., v. 29, 1975, pp. 978-984. MR 0388809 (52:9643)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N30, 65M15

Retrieve articles in all journals with MSC: 65N30, 65M15


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1981-0595043-8
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society