Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 

 

Application of method of collocation on lines for solving nonlinear hyperbolic problems


Author: E. N. Houstis
Journal: Math. Comp. 31 (1977), 443-456
MSC: Primary 65N35
MathSciNet review: 0443379
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A collocation on lines procedure based on piecewise polynomials is applied to initial/boundary value problems for nonlinear hyperbolic partial differential equations. Optimal order a priori estimates are obtained for the error of approximation. The Crank-Nicholson discretization in time is studied and convergence rates of the collocation-Crank-Nicholson procedure are established. Finally, the superconvergence is verified at particular points for linear hyperbolic problems.


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

  • [1] I. S. Berezin and N. P. Shidkov, Computing methods. Vols. I, II, Translated by O. M. Blunn; translation edited by A. D. Booth, Pergamon Press, Oxford-Edinburgh-New York-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1965. Vol. I: xxxiv+464 pp. $ 15.00; Vol. II, 1965. MR 0174165
  • [2] Earl A. Coddington, An introduction to ordinary differential equations, Prentice-Hall Mathematics Series, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1961. MR 0126573
  • [3] Jim Douglas Jr. and Todd Dupont, A finite element collocation method for quasilinear parabolic equations, Math. Comp. 27 (1973), 17–28. MR 0339508, 10.1090/S0025-5718-1973-0339508-8
  • [4] Jim Douglas Jr. and Todd Dupont, A finite element collocation method for the heat equation, Symposia Mathematica, Vol. X (Convegno di Analisi Numerica, INDAM, Rome, 1972) Academic Press, London, 1972, pp. 403–410. MR 0373329
  • [5] Jim Douglas Jr. and Todd Dupont, Collocation methods for parabolic equations in a single space variable, Lecture Notes in Mathematics, Vol. 385, Springer-Verlag, Berlin-New York, 1974. Based on 𝐶¹-piecewise-polynomial spaces. MR 0483559
  • [6] E. N. HOUSTIS, Finite Element Methods for Solving Initial/Boundary Value Problems, Doctoral thesis, Purdue University, 1974.
  • [7] L. V. KANTOROVIČ, "Sur une méthode de resolution approchée d'equations différentielles aux derivées partielles," C. R. Acad. (Dokl.) Sci. URSS , v. 2, 1934, pp. 532-536. (Russian)
  • [8] È. B. Karpilovskaja, Convergence of a collocation method for certain boundary-value problems of mathematical physics, Sibirsk. Mat. Ž. 4 (1963), 632–640 (Russian). MR 0156479
  • [9] Martin H. Schultz, Spline analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. Prentice-Hall Series in Automatic Computation. MR 0362832
  • [10] Ju. P. Jarcev, The convergence of the method of collocation by lines, Differencial′nye Uravnenija 3 (1967), 1606–1613 (Russian). MR 0221768
  • [11] Yu. P. YARTSEV, "The method of line collocation," Differencial'nye Uravnenija, v. 4, 1968, pp. 925-932 = Differential Equations, v. 4, 1968, pp. 481-485.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N35

Retrieve articles in all journals with MSC: 65N35


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1977-0443379-2
Keywords: Collocation on lines method, nonlinear hyperbolic problems
Article copyright: © Copyright 1977 American Mathematical Society