Estimates away from a discontinuity for dissipative Galerkin methods for hyperbolic equations
Author:
William J. Layton
Journal:
Math. Comp. 36 (1981), 8792
MSC:
Primary 65N30; Secondary 65M15
MathSciNet review:
595043
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Abstract: We consider the approximate solution of the initial value problem by a dissipative Galerkin method. When v is taken to have a jump discontinuity at zero, that discontinuity will propagate along , in the true solution u. Estimates in and of the pollution effects of the discontinuity are found. These estimates show those effects to decay exponentially in in regions a fixed distance d from the discontinuity and exponentially in d for fixed h.
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 [1]
 M. Y. T. Apelkrans, "On difference schemes for hyperbolic equations with discontinuous initial values," Math. Comp., v. 22, 1968, pp. 525539. MR 0233527 (38:1848)
 [2]
 Ph. Brenner & V. Thomée, "Stability and convergence rates in , for certain difference schemes," Math. Scand., v. 27, 1970, pp. 523. MR 0278549 (43:4279)
 [3]
 Ph. Brenner & V. ThomÉe, "Estimates near discontinuities for some difference schemes," Math. Scand., v. 28, 1971, pp. 329340. MR 0305613 (46:4743)
 [4]
 A. Calderón, F. Spitzer & H. Widom, "Inversion of Toeplitz matrices," Illinois J. Math., v. 3, 1959, pp. 490498. MR 0121652 (22:12386)
 [5]
 J. E. Dendy, "Two methods of Galerkin type achieving optimum rates of convergence for first order hyperbolics," SIAM J. Numer. Anal., v. 11, 1974, pp. 637653. MR 0353695 (50:6178)
 [6]
 G. W. Hedstrom, "The rate of convergence of some difference schemes," SIAM J. Numer. Anal., v. 5, 1968, pp. 363406. MR 0230489 (37:6051)
 [7]
 W. J. Layton, Ph. D. Thesis, University of Tennessee, 1980.
 [8]
 R. Richards, Uniform Spline Interpolation Operators in , 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. 404420. 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. 411424.
 [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. 10591068. 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. 109117. MR 0368447 (51:4688)
 [19]
 L. B. Wahlbin, "A modified Galerkin procedure with Hermite cubics for hyperbolic problems," Math. Comp., v. 29, 1975, pp. 978984. MR 0388809 (52:9643)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198105950438
PII:
S 00255718(1981)05950438
Article copyright:
© Copyright 1981
American Mathematical Society
