A simplified Galerkin method for hyperbolic equations
HTML articles powered by AMS MathViewer
- by R. C. Y. Chin, G. W. Hedstrom and K. E. Karlsson PDF
- Math. Comp. 33 (1979), 647-658 Request permission
Abstract:
We modify a Galerkin method for nonlinear hyperbolic equations so that it becomes a simpler method of lines, which may be viewed as a collocation method. The high order of accuracy is preserved. We present a linear wave analysis of the scheme and discuss some aspects of nonlinear problems. Our numerical experiments indicate that the addition of a proper artificial viscosity makes the method competitive and the common difference schemes, even when the solution has discontinuities.References
- Approximation by hill functions, Comment. Math. Univ. Carolinae 11 (1970), 787–811. MR 292309
- Edward R. Benton and George W. Platzman, A table of solutions of the one-dimensional Burgers equation, Quart. Appl. Math. 30 (1972), 195–212. MR 306736, DOI 10.1090/S0033-569X-1972-0306736-4
- Léon Brillouin, Wave propagation and group velocity, Pure and Applied Physics, Vol. 8, Academic Press, New York-London, 1960. MR 0108217 T. J. BROMWICH, An Introduction to the Theory of Infinite Series, Macmillan, London, 1949.
- R. C. Y. Chin and G. W. Hedstrom, A dispersion analysis for difference schemes: tables of generalized Airy functions, Math. Comp. 32 (1978), no. 144, 1163–1170. MR 494982, DOI 10.1090/S0025-5718-1978-0494982-6
- J. Albrecht and L. Collatz (eds.), Numerische Behandlung von Differentialgleichungen. Band 3, Internationale Schriftenreihe zur Numerischen Mathematik [International Series of Numerical Mathematics], vol. 56, Birkhäuser Verlag, Basel, 1981 (German). MR 784038, DOI 10.1007/978-3-0348-5454-2
- B. Fornberg, On the instability of leap-frog and Crank-Nicolson approximations of a nonlinear partial differential equation, Math. Comp. 27 (1973), 45–57. MR 395249, DOI 10.1090/S0025-5718-1973-0395249-2
- C. William Gear, Numerical initial value problems in ordinary differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0315898
- James Glimm and Peter D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Memoirs of the American Mathematical Society, No. 101, American Mathematical Society, Providence, R.I., 1970. MR 0265767 A. C. HINDMARSH, GEAR: Ordinary Differential Equation System Solver, Lawrence Livermore Laboratory Report UCID-30001, Rev. 1. A.C.C. No. 592, GEAR. Argonne Code Center, Building 221, Argonne National Laboratory, Argonne, Illinois.
- Leon Lapidus and John H. Seinfeld, Numerical solution of ordinary differential equations, Mathematics in Science and Engineering, Vol. 74, Academic Press, New York-London, 1971. MR 0281355 R. D. RICHTMEYER & K. W. MORTON, Difference Methods for Initial-Value Problems, 2nd ed., Interscience, New York, 1967. MR 36 #3515. I. J. SCHOENBERG, "Contributions to the approximation of equidistant data by analytic functions," Quart. Appl. Math., v. 4, 1946, pp. 45-99 and 112-141. MR 7, 487 and 8, 55.
- S. I. Serdjukova, The oscillations that arise in numerical calculations of the discontinuous solutions of differential equations, Ž. Vyčisl. Mat i Mat. Fiz. 11 (1971), 411–424 (Russian). MR 284018
- Gilbert Strang, The finite element method and approximation theory, Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970) Academic Press, New York, 1971, pp. 547–583. MR 0287723
- Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0443377
- Blair Swartz and Burton Wendroff, Generalized finite-difference schemes, Math. Comp. 23 (1969), 37–49. MR 239768, DOI 10.1090/S0025-5718-1969-0239768-7
- Blair Swartz and Burton Wendroff, The relation between the Galerkin and collocation methods using smooth splines, SIAM J. Numer. Anal. 11 (1974), 994–996. MR 362953, DOI 10.1137/0711077
- Blair Swartz and Burton Wendroff, The relative efficiency of finite difference and finite element methods. I. Hyperbolic problems and splines, SIAM J. Numer. Anal. 11 (1974), 979–993. MR 362952, DOI 10.1137/0711076
- E. L. Allgower and M. M. Jeppson, The approximation of solutions of nonlinear elliptic boundary value problems having several solutions, Numerische, insbesondere approximationstheoretische Behandlung von Funktionalgleichungen (Tagung, Math. Forschungsinst., Oberwolfach, 1972) Lecture Notes in Math., Vol. 333, Springer, Berlin, 1973, pp. 1–20. MR 0433923
- 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. VICHNEVETSKY & B. PEIFFER, "Error waves in finite element and finite difference methods for hyperbolic equations," in Advances in Computer Methods for Partial Differential Equations, R. Vichnevetsky, (ed.), Assoc. Int. Calcul Analogique, Ghent, Belgium, 1975, pp. 1-6. R. VICHNEVETSKY & F. DE SHUTTER, "A frequency analysis of finite difference and finite element methods for initial-value problems," in Advances in Computer Methods for Partial Differential Equations, R. Vichnevetsky, (ed.), Assoc. Int. Calcul Analogique, Ghent, Belgium, 1975, pp. 46-52.
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Math. Comp. 33 (1979), 647-658
- MSC: Primary 65M10; Secondary 65N30
- DOI: https://doi.org/10.1090/S0025-5718-1979-0521280-5
- MathSciNet review: 521280