Estimates away from a discontinuity for dissipative Galerkin methods for hyperbolic equations
HTML articles powered by AMS MathViewer
- by William J. Layton PDF
- Math. Comp. 36 (1981), 87-92 Request permission
Abstract:
We consider the approximate solution of the initial value problem \[ \frac {{\partial u}}{{\partial t}} = \frac {{\partial u}}{{\partial x}},\quad u(x,0) = v(x),\] 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
- Mats Y. T. Apelkrans, On difference schemes for hyperbolic equations with discontinuous initial values, Math. Comp. 22 (1968), 525–539. MR 233527, DOI 10.1090/S0025-5718-1968-0233527-6
- Philip Brenner and Vidar Thomée, Stability and convergence rates in $L_{p}$ for certain difference schemes, Math. Scand. 27 (1970), 5–23. MR 278549, DOI 10.7146/math.scand.a-10983
- Philip Brenner and Vidar Thomée, Estimates near discontinuities for some difference schemes, Math. Scand. 28 (1971), 329–340 (1972). MR 305613, DOI 10.7146/math.scand.a-11028
- Alberto Calderón, Frank Spitzer, and Harold Widom, Inversion of Toeplitz matrices, Illinois J. Math. 3 (1959), 490–498. MR 121652
- J. E. Dendy, Two methods of Galerkin type achieving optimum $L^{2}$ rates of convergence for first order hyperbolics, SIAM J. Numer. Anal. 11 (1974), 637–653. MR 353695, DOI 10.1137/0711052
- G. W. Hedstrom, The rate of convergence of some difference schemes, SIAM J. Numer. Anal. 5 (1968), 363–406. MR 230489, DOI 10.1137/0705031 W. J. Layton, Ph. D. Thesis, University of Tennessee, 1980. R. Richards, Uniform Spline Interpolation Operators in ${L_2}$, MRC Tech. Report # 1305, University of Wisconsin, November 1972.
- Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, 2nd ed., Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0220455
- I. J. Schoenberg, Cardinal interpolation and spline functions. II. Interpolation of data of power growth, J. Approximation Theory 6 (1972), 404–420. MR 340899, DOI 10.1016/0021-9045(72)90048-2
- I. J. Schoenberg, Cardinal spline interpolation, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 12, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. MR 0420078 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.
- Vidar Thomée, Spline approximation and difference schemes for the heat equation, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 711–746. MR 0403265
- Vidar Thomée, Convergence estimates for semi-discrete Galerkin methods for initial-value problems, Numerische, insbesondere approximationstheoretische Behandlung von Funktionalgleichungen (Tagung, Math. Forschungsinst., Oberwolfach, 1972) Lecture Notes in Math., Vol. 333, Springer, Berlin, 1973, pp. 243–262. MR 0458948
- Vidar Thomée and Burton Wendroff, Convergence estimates for Galerkin methods for variable coefficient initial value problems, SIAM J. Numer. Anal. 11 (1974), 1059–1068. MR 371088, DOI 10.1137/0711081
- R. S. Varga, Functional analysis and approximation theory in numerical analysis, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 3, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1971. MR 0310504
- Lars B. Wahlbin, A dissipative Galerkin method for the numerical solution of first order hyperbolic equations, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974) Publication No. 33, Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 147–169. MR 0658322
- Lars B. Wahlbin, A dissipative Galerkin method applied to some quasilinear hyperbolic equations, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 109–117 (English, with French summary). MR 368447
- Lars Wahlbin, A modified Galerkin procedure with Hermite cubics for hyperbolic problems, Math. Comput. 29 (1975), no. 132, 978–984. MR 0388809, DOI 10.1090/S0025-5718-1975-0388809-8
Additional Information
- © Copyright 1981 American Mathematical Society
- 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